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Problems in Geometry

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Problem 1 [BMOTC]<br />

<strong>Problems</strong> <strong>in</strong> <strong>Geometry</strong><br />

Prithwijit De<br />

ICFAI Bus<strong>in</strong>ess School, Kolkata<br />

Republic of India<br />

email: de.prithwijit@gmail.com<br />

Prove that the medians from the vertices A and B of triangle ABC are<br />

mutually perpendicular if and only if |BC| 2 + |AC| 2 = 5|AB| 2 .<br />

Problem 2 [BMOTC]<br />

Suppose that ∠A is the smallest of the three angles of triangle ABC. Let D<br />

be a po<strong>in</strong>t on the arc BC of the circumcircle of ABC which does not conta<strong>in</strong><br />

A. Let the perpendicular bisectors of AB, AC <strong>in</strong>tersect AD at M and N<br />

respectively. Let BM and CN meet at T . Prove that BT + CT ≤ 2R where<br />

R is the circumradius of triangle ABC.<br />

Problem 3 [BMOTC]<br />

Let triangle ABC have side lengths a, b and c as usual. Po<strong>in</strong>ts P and Q<br />

lie <strong>in</strong>side this triangle and have the properties that ∠BP C = ∠CP A =<br />

∠AP B = 120 ◦ and ∠BQC = 60 ◦ + ∠A, ∠CQA = 60 ◦ + ∠B, ∠AQB =<br />

60 ◦ + ∠C. Prove that<br />

Problem 4 [BMOTC]<br />

(|AP | + |BP | + |CP |) 3 .|AQ|.|BQ|.|CQ| = (abc) 2 .<br />

The po<strong>in</strong>ts M and N are the po<strong>in</strong>ts of tangency of the <strong>in</strong>circle of the isosceles<br />

triangle ABC which are on the sides AC and BC. The sides of equal length<br />

are AC and BC. A tangent l<strong>in</strong>e t is drawn to the m<strong>in</strong>or arc MN. Suppose<br />

that t <strong>in</strong>tersects AC and BC at Q and P respectively. Suppose that the l<strong>in</strong>es<br />

AP and BQ meet at T .<br />

(a) Prove that T lies on the l<strong>in</strong>e segment MN.<br />

(b) Prove that the sum of the areas of triangles AT Q and BT P is<br />

m<strong>in</strong>imized when t is parallel to AB.<br />

Problem 5 [BMOTC]<br />

In a hexagon with equal angles, the lengths of four consecutive edges are 5,<br />

3, 6 and 7 (<strong>in</strong> that order). F<strong>in</strong>d the lengths of the rema<strong>in</strong><strong>in</strong>g two edges.<br />

1


Problem 6 [BMOTC]<br />

The <strong>in</strong>circle γ of triangle ABC touches the side AB at T . Let D be the po<strong>in</strong>t<br />

on γ diametrically opposite to T , and let S be the <strong>in</strong>tersection of the l<strong>in</strong>e<br />

through C and D with the side AB. Show that |AT | = |SB|.<br />

Problem 7 [BMOTC]<br />

Let S and r be the area and the <strong>in</strong>radius of the triangle ABC. Let r A denote<br />

the radius of the circle touch<strong>in</strong>g the <strong>in</strong>circle, AB and AC. Def<strong>in</strong>e r B and<br />

r C similarly. The common tangent of the circles with radii r and r A cuts a<br />

little triangle from ABC with area S A . Quantities S B and S C are def<strong>in</strong>ed <strong>in</strong><br />

a similar fashion. Prove that<br />

S A<br />

rA<br />

+ S B<br />

rB<br />

+ S C<br />

rC<br />

= S r<br />

Problem 8 [BMOTC]<br />

Triangle ABC <strong>in</strong> the plane Π is said to be good if it has the follow<strong>in</strong>g property:<br />

for any po<strong>in</strong>t D <strong>in</strong> space, out of the plane Π, it is possible to construct a<br />

triangle with sides of lengths |AD|, |BD| and |CD|. F<strong>in</strong>d all good triangles.<br />

Problem 9 [BMO]<br />

Circle γ lies <strong>in</strong>side circle θ and touches it at A. From a po<strong>in</strong>t P (dist<strong>in</strong>ct<br />

from A) on θ, chords P Q and P R of θ are drawn touch<strong>in</strong>g γ at X and Y<br />

respectively. Show that ∠QAR = 2∠XAY .<br />

Problem 10 [BMO]<br />

AP , AQ, AR, AS are chords of a given circle with the property that<br />

Prove that<br />

Problem 11 [BMO]<br />

∠P AQ = ∠QAR = ∠RAS.<br />

AR(AP + AR) = AQ(AQ + AS).<br />

The po<strong>in</strong>ts Q, R lie on the circle γ, and P is a po<strong>in</strong>t such that P Q, P R are<br />

tangents to γ. A is a po<strong>in</strong>t on the extension of P Q and γ ′ is the circumcircle<br />

of triangle P AR. The circle γ ′ cuts γ aga<strong>in</strong> at B and AR cuts γ at the po<strong>in</strong>t<br />

C. Prove that ∠P AR = ∠ABC.<br />

2


Problem 12 [BMO]<br />

In the acute-angled triangle ABC, CF is an altitude, with F on AB and BM<br />

is a median with M on CA. Given that BM = CF and ∠MBC = ∠F CA,<br />

prove that the triangle ABC is equilateral.<br />

Problem 13 [BMO]<br />

A triangle ABC has ∠BAC > ∠BCA. A l<strong>in</strong>e AP is drawn so that ∠P AC =<br />

∠BCA where P is <strong>in</strong>side the triangle. A po<strong>in</strong>t Q outside the triangle is<br />

constructed so that P Q is parallel to AB, and BQ is parallel to AC. R is the<br />

po<strong>in</strong>t on BC (separated from Q by the l<strong>in</strong>e AP ) such that ∠P RQ = ∠BCA.<br />

Prove that the circumcircle of ABC touches the circumcircle of P QR.<br />

Problem 14 [BMO]<br />

ABP is an isosceles triangle with AB=AP and ∠P AB acute. P C is the<br />

l<strong>in</strong>e through P perpendicular to BP and C is a po<strong>in</strong>t on this l<strong>in</strong>e on the<br />

same side of BP as A. (You may assume that C is not on the l<strong>in</strong>e AB). D<br />

completes the parallelogram ABCD. P C meets DA at M. Prove that M is<br />

the midpo<strong>in</strong>t of DA.<br />

Problem 15 [BMO]<br />

In triangle ABC, D is the midpo<strong>in</strong>t of AB and E is the po<strong>in</strong>t of trisection<br />

of BC nearer to C. Given that ∠ADC = ∠BAE f<strong>in</strong>d ∠BAC.<br />

Problem 16 [BMO]<br />

ABCD is a rectangle, P is the midpo<strong>in</strong>t of AB and Q is the po<strong>in</strong>t on P D<br />

such that CQ is perpendicular to P D. Prove that BQC is isosceles.<br />

Problem 17 [BMO]<br />

Let ABC be an equilateral triangle and D an <strong>in</strong>ternal po<strong>in</strong>t of the side BC.<br />

A circle, tangent to BC at D, cuts AB <strong>in</strong>ternally at M and N and AC<br />

<strong>in</strong>ternally at P and Q. Show that BD + AM + AN = CD + AP + AQ.<br />

Problem 18 [BMO]<br />

Let ABC be an acute-angled triangle, and let D, E be the feet of the perpendiculars<br />

from A, B to BC and CA respectively. Let P be the po<strong>in</strong>t where<br />

the l<strong>in</strong>e AD meets the semicircle constructed outwardly on BC and Q be the<br />

po<strong>in</strong>t where the l<strong>in</strong>e BE meets the semicircle constructed outwardly on AC.<br />

Prove that CP = CQ.<br />

3


Problem 19 [BMO]<br />

Two <strong>in</strong>tersect<strong>in</strong>g circles C 1 and C 2 have a common tangent which touches<br />

C 1 at P and C 2 at Q. The two circles <strong>in</strong>tersect at M and N, where N is<br />

closer to P Q than M is. Prove that the triangles MNP and MNQ have<br />

equal areas.<br />

Problem 20 [BMO]<br />

Two <strong>in</strong>tersect<strong>in</strong>g circles C 1 and C 2 have a common tangent which touches C 1<br />

at P and C 2 at Q. The two circles <strong>in</strong>tersect at M and N, where N is closer<br />

to P Q than M is. The l<strong>in</strong>e P N meets the circle C 2 aga<strong>in</strong> at R. Prove that<br />

MQ bisects ∠P MR.<br />

Problem 21 [BMO]<br />

Triangle ABC has a right angle at A. Among all po<strong>in</strong>ts P on the perimeter<br />

of the triangle, f<strong>in</strong>d the position of P such that AP +BP +CP is m<strong>in</strong>imized.<br />

Problem 22 [BMO]<br />

Let ABCDEF be a hexagon (which may not be regular), which circumscribes<br />

a circle S. (That is, S is tangent to each of the six sides of the hexagon.)<br />

The circle S touches AB, CD, EF at their midpo<strong>in</strong>ts P , Q, R respectively.<br />

Let X, Y , Z be the po<strong>in</strong>ts of contact of S with BC, DE, F A respectively.<br />

Prove that P Y , QZ, RX are concurrent.<br />

Problem 23 [BMO]<br />

The quadrilateral ABCD is <strong>in</strong>scribed <strong>in</strong> a circle. The diagonals AC, BD<br />

meet at Q. The sides DA, extended beyond A, and CB, extended beyond<br />

B, meet at P . Given that CD = CP = DQ, prove that ∠CAD = 60 ◦ .<br />

Problem 24 [BMO]<br />

The sides a, b, c and u, v, w of two triangles ABC and UV W are related by<br />

the equations<br />

u(v + w − u) = a 2<br />

v(w + u − v) = b 2<br />

w(u + v − w) = c 2<br />

Prove that triangle ABC is acute-angled and express the angles U, V , W <strong>in</strong><br />

terms of A, B, C.<br />

4


Problem 25 [BMO]<br />

Two circles S 1 and S 2 touch each other externally at K; they also touch a<br />

circle S <strong>in</strong>ternally at A 1 and A 2 respectively. Let P be one po<strong>in</strong>t of <strong>in</strong>tersection<br />

of S with the common tangent to S 1 and S 2 at K. The l<strong>in</strong>e P A 1 meets<br />

S 1 aga<strong>in</strong> at B 1 and P A 2 meets S 2 aga<strong>in</strong> at B 2 . Prove that B 1 B 2 is a common<br />

tangent to S 1 and S 2 .<br />

Problem 26 [BMO]<br />

Let ABC be an acute-angled triangle and let O be its circumcentre. The<br />

circle through A, O and B is called S. The l<strong>in</strong>es CA and CB meet the<br />

circle S aga<strong>in</strong> at P and Q respectively. Prove that the l<strong>in</strong>es CO and P Q are<br />

perpendicular.<br />

Problem 27 [BMO]<br />

Two circles touch <strong>in</strong>ternally at M. A straight l<strong>in</strong>e touches the <strong>in</strong>ner circle at<br />

P and cuts the outer circle at Q and R. Prove that ∠QMP = ∠RMP .<br />

Problem 28 [BMO]<br />

ABC is a triangle, right-angled at C. The <strong>in</strong>ternal bisectors of ∠BAC and<br />

∠ABC meet BC and CA at P and Q, respectively. M and N are the feet<br />

of the perpendiculars from P and Q to AB. F<strong>in</strong>d the measure of ∠MCN.<br />

Problem 29 [BMO]<br />

The triangle ABC, where AB < AC, has circumcircle S. The perpendicular<br />

from A to BC meets S aga<strong>in</strong> at P . The po<strong>in</strong>t X lies on the segment AC<br />

and BX meets S aga<strong>in</strong> at Q. Show that BX = CX if and only if P Q is a<br />

diameter of S.<br />

Problem 30 [BMO]<br />

Let ABC be a triangle and let D be a po<strong>in</strong>t on AB such that 4AD = AB.<br />

The half-l<strong>in</strong>e l is drawn on the same side of AB as C, start<strong>in</strong>g from D and<br />

mak<strong>in</strong>g an angle of θ with DA where θ = ∠ACB. If the circumcircle of ABC<br />

meets the half-l<strong>in</strong>e l at P , show that P B = 2P D.<br />

5


Problem 31 [BMO]<br />

Let BE and CF be the altitudes of an acute triangle ABC, with E on AC<br />

and F on AB. Let O be the po<strong>in</strong>t of <strong>in</strong>tersection of BE and CF . Take any<br />

l<strong>in</strong>e KL through O with K on AB and L on AC. Suppose M and N are<br />

located on BE and CF respectively, such that KM is perpendicular to BE<br />

and LN is perpendicular to CF . Prove that F M is parallel to EN.<br />

Problem 32 [BMO]<br />

In a triangle ABC, D is a po<strong>in</strong>t on BC such that AD is the <strong>in</strong>ternal bisector<br />

of ∠A. Suppose ∠B = 2∠C and CD = AB. Prove that ∠A = 72 ◦ .<br />

Problem 33 [Putnam]<br />

Let T be an acute triangle. Inscribe a rectangle R <strong>in</strong> T with one side along<br />

a side of T . Then <strong>in</strong>scribe a rectangle S <strong>in</strong> the triangle formed by the side<br />

of R opposite the side on the boundary of T , and the other two sides of T ,<br />

with one side along the side of R. For any polygon X, let A(X) denote the<br />

area of X. F<strong>in</strong>d the maximum value, or show that no maximum exists, of<br />

A(R)+A(S)<br />

where T ranges over all triangles and R, S over all rectangles as<br />

A(T )<br />

above.<br />

Problem 34 [Putnam]<br />

A rectangle, HOMF , has sides HO=11 and OM=5. A triangle ABC has<br />

H as the orthocentre, O as the circumcentre, M the midpo<strong>in</strong>t of BC and F<br />

the foot of the altitude from A. What is the length of BC?<br />

Problem 35 [Putnam]<br />

A right circular cone has base of radius 1 and height 3. A cube is <strong>in</strong>scribed<br />

<strong>in</strong> the cone so that one face of the cube is conta<strong>in</strong>ed <strong>in</strong> the base of the cone.<br />

What is the side-length of the cube?<br />

Problem 36 [Putnam]<br />

Let A, B and C denote dist<strong>in</strong>ct po<strong>in</strong>ts with <strong>in</strong>teger coord<strong>in</strong>ates <strong>in</strong> R 2 . Prove<br />

that if (|AB| + |BC|) 2 < 8[ABC] + 1 then A, B, C are three vertices of a<br />

square. Here |XY | is the length of segment XY and [ABC] is the area of<br />

triangle ABC.<br />

6


Problem 37 [Putnam]<br />

Right triangle ABC has right angle at C and ∠BAC = θ; the po<strong>in</strong>t D is<br />

chosen on AB so that |AC| = |AD| = 1; the po<strong>in</strong>t E is chosen on BC so<br />

that ∠CDE = θ. The perpendicular to BC at E meets AB at F . Evaluate<br />

lim θ→0 |EF |.<br />

Problem 38 [BMO]<br />

Let ABC be a triangle and D, E, F be the midpo<strong>in</strong>ts of BC, CA, AB<br />

respectively. Prove that ∠DAC = ∠ABE if, and only if, ∠AF C = ∠ADB.<br />

Problem 39 [BMO]<br />

The altitude from one of the vertex of an acute-angled triangle ABC meets<br />

the opposite side at D. From D perpendiculars DE and DF are drawn to the<br />

other two sides. Prove that the length of EF is the same whichever vertex<br />

is chosen.<br />

Problem 40<br />

Two cyclists ride round two <strong>in</strong>tersect<strong>in</strong>g circles, each mov<strong>in</strong>g with a constant<br />

speed. Hav<strong>in</strong>g started simultaneously from a po<strong>in</strong>t at which the circles <strong>in</strong>tersect,<br />

the cyclists meet once aga<strong>in</strong> at this po<strong>in</strong>t after one circuit. Prove<br />

that there is a fixed po<strong>in</strong>t such that the distances from it to the cyclists are<br />

equal all the time if they ride: (a) <strong>in</strong> the same direction (clockwise); (b) <strong>in</strong><br />

opposite direction.<br />

Problem 41<br />

Prove that four circles circumscribed about four triangles formed by four<br />

<strong>in</strong>tersect<strong>in</strong>g straight l<strong>in</strong>es <strong>in</strong> the plane have a common po<strong>in</strong>t. (Michell’s<br />

Po<strong>in</strong>t).<br />

Problem 42<br />

Given an equilateral triangle ABC. F<strong>in</strong>d the locus of po<strong>in</strong>ts M <strong>in</strong>side the<br />

triangle such that ∠MAB + ∠MBC + ∠MCA = π 2 .<br />

Problem 43<br />

In a triangle ABC, on the sides AC and BC, po<strong>in</strong>ts M and N are taken,<br />

respectively and a po<strong>in</strong>t L on the l<strong>in</strong>e segment MN. Let the areas of the<br />

triangles ABC, AML and BNL be equal to S, P and Q, respectively. Prove<br />

that<br />

7


S 1 3 ≥ P 1 3 + Q 1 3 .<br />

Problem 44<br />

bc cos A<br />

For an arbitrary triangle, prove the <strong>in</strong>equality + a < p < bc+a2 , where<br />

b+c<br />

a<br />

a, b and c are the sides of the triangle and p its semiperimeter.<br />

Problem 45<br />

Given <strong>in</strong> a triangle are two sides: a and b (a > b). F<strong>in</strong>d the third side if it is<br />

known that a + h a ≤ b + h b , where h a and h b are the altitudes dropped on<br />

these sides (h a the altitude drawn to the side a).<br />

Problem 46<br />

One of the sides <strong>in</strong> a triangle ABC is twice the length of the other and<br />

∠B = 2∠C. F<strong>in</strong>d the angles of the triangle.<br />

Problem 47<br />

In a parallelogram whose area is S, the bisectors of its <strong>in</strong>terior angles are<br />

drawn to <strong>in</strong>tersect one another. The area of the quadrilateral thus obta<strong>in</strong>ed<br />

is equal to Q. F<strong>in</strong>d the ratio of the sides of the parallelogram.<br />

Problem 48<br />

Prove that if one angle of a triangle is equal to 120 ◦ , then the triangle formed<br />

by the feet of its angle bisectors is right-angled.<br />

Problem 49<br />

Given a rectangle ABCD where |AB| = 2a, |BC| = a √ 2. With AB is<br />

diameter a semicircle is constructed externally. Let M be an arbitrary po<strong>in</strong>t<br />

on the semicircle, the l<strong>in</strong>e MD <strong>in</strong>tersect AB at N, and the l<strong>in</strong>e MC at L.<br />

F<strong>in</strong>d |AL| 2 + |BN| 2 .<br />

Problem 50<br />

Let A, B and C be three po<strong>in</strong>ts ly<strong>in</strong>g on the same l<strong>in</strong>e. Constructed on AB,<br />

BC and AC as diameters are three semicircles located on the same side of<br />

the l<strong>in</strong>e. The centre of a circle touch<strong>in</strong>g the three semicircles is found at a<br />

distance d from the l<strong>in</strong>e AC. F<strong>in</strong>d the radius of this circle.<br />

8


Problem 51<br />

In an isosceles triangle ABC, |AC| = |BC|, BD is an angle bisector, BDEF<br />

is a rectangle. F<strong>in</strong>d ∠BAF if ∠BAE = 120 ◦ .<br />

Problem 52<br />

Let M 1 be a po<strong>in</strong>t on the <strong>in</strong>circle of triangle ABC. The perpendiculars to<br />

the sides through M 1 meet the <strong>in</strong>circle aga<strong>in</strong> at M 2 , M 3 , M 4 . Prove that the<br />

geometric mean of the six lengths M i M j , 1 ≤ i ≤ j ≤ 4, is less than or equal<br />

to r 3√ 4, where r denotes the <strong>in</strong>radius. When does the equality hold?<br />

Problem 53 [AMM]<br />

Let ABC be a triangle and let I be the <strong>in</strong>circle of ABC and let r be the<br />

radius of I. Let K 1 , K 2 and K 3 be the three circles outside I and tangent<br />

to I and to two of the three sides of ABC. Let r i be the radius of K i for<br />

1 ≤ i ≤ 3. Show that<br />

r = √ r 1 r 2 + √ r 2 r 3 + √ r 3 r 1<br />

Problem 54 [Prithwijit’s Inequality]<br />

In triangle ABC suppose the lengths of the medians are m a , m b and m c<br />

respectively. Prove that<br />

Problem 55 [Loney]<br />

am a+bm b +cm c<br />

(a+b+c)(m a+m b +m c) ≤ 1 3<br />

The base a of a triangle and the ratio r(< 1) of the sides are given. Show<br />

that the altitude h of the triangle cannot exceed<br />

ar and that when h has<br />

1−r 2<br />

this value the vertical angle of the triangle is π − 2 2 tan−1 r.<br />

Problem 56 [Loney]<br />

The <strong>in</strong>ternal bisectors of the angles of a triangle ABC meet the sides <strong>in</strong> D,<br />

E and F . Show that the area of the triangle DEF is equal to<br />

Problem 57 [Loney]<br />

2∆abc<br />

.<br />

(a+b)(b+c)(c+a)<br />

If a, b, c are the sides of a triangle, λa, λb, λc the sides of a similar triangle<br />

<strong>in</strong>scribed <strong>in</strong> the former and θ the angle between the sides a and λa, prove<br />

that 2λ cos θ = 1.<br />

9


Problem 58<br />

Let a, b and c denote the sides of a triangle and a + b + c = 2p. Let G be the<br />

median po<strong>in</strong>t of the triangle and O, I and I a the centres of the circumscribed,<br />

<strong>in</strong>scribed and escribed circles, respectively (the escribed circle touches the<br />

side BC and the extensions of the sides AB and AC), R, r and r a be<strong>in</strong>g<br />

their radii, respectively. Prove that the follow<strong>in</strong>g relationships are valid:<br />

(a) a 2 + b 2 + c 2 = 2p 2 − 2r 2 − 8Rr<br />

(b) |OG| 2 = R 2 − a2 +b 2 +c 2<br />

9<br />

(c) |IG| 2 = p2 +5r 2 −16Rr<br />

9<br />

(d) |OI| 2 = R 2 − 2Rr<br />

(e) |OI a | 2 = R 2 + 2Rr a<br />

(f) |II a | 2 = 4R(r a − r)<br />

Problem 59<br />

MN is a diameter of a circle, |MN| = 1, A and B are po<strong>in</strong>ts on the circles<br />

situated on one side of MN, C is a po<strong>in</strong>t on the other semicircle. Given: A<br />

is the midpo<strong>in</strong>t of semicircle, MB = 3 , the length of the l<strong>in</strong>e segment formed<br />

5<br />

by the <strong>in</strong>tersection of the diameter MN with the chords AC and BC is equal<br />

to a. What is the greatest value of a?<br />

Problem 60<br />

Given a parallelogram ABCD. A straight l<strong>in</strong>e pass<strong>in</strong>g through the vertex C<br />

<strong>in</strong>tersects the l<strong>in</strong>es AB and AD at po<strong>in</strong>ts K and L, respectively. The areas<br />

of the triangles KBC and CDL are equal to p and q, respectively. F<strong>in</strong>d the<br />

area of the parallelogram ABCD.<br />

Problem 61 [Loney]<br />

Three circles, whose radii are a, b and c, touch one another externally and the<br />

tangents at their po<strong>in</strong>ts of contact meet <strong>in</strong> a po<strong>in</strong>t; prove √ that the distance<br />

of this po<strong>in</strong>t from either of their po<strong>in</strong>ts of contact is<br />

Problem 62 [Loney]<br />

abc<br />

. a+b+c<br />

If a circle be drawn touch<strong>in</strong>g the <strong>in</strong>scribed and circumscribed circles of a<br />

triangle and the side BC externally, prove that its radius is ∆ a tan2 A 2 .<br />

10


Problem 63<br />

Characterize all triangles ABC such that<br />

AI a : BI b : CI c = BC : CA : AB<br />

where I a ; I b , I c are the vertices of the excentres correspond<strong>in</strong>g to A, B, C<br />

respectively.<br />

Problem 64<br />

On the sides AB and BC of triangle ABC, po<strong>in</strong>ts K and M are chosen such<br />

that the quadrilaterals AKMC and KBMN are cyclic, where<br />

N = AM ∩ CK. If these quadrilaterals have the same circumradii then f<strong>in</strong>d<br />

∠ABC.<br />

Problem 65 [AMM]<br />

Let B ′ and C ′ be po<strong>in</strong>ts on the sides AB and AC, respectively, of a given<br />

triangle ABC, and let P be a po<strong>in</strong>t on the segment B ′ C ′ . Determ<strong>in</strong>e the<br />

maximum value of<br />

where [F ] denotes the area of F .<br />

Problem 66 [AMM]<br />

m<strong>in</strong>([BP B ′ ],[CP C ′ ])<br />

[ABC]<br />

For each po<strong>in</strong>t O on diameter AB of a circle, perform the follow<strong>in</strong>g construction.<br />

Let the perpendicular to AB at O meet the circle at po<strong>in</strong>t P . Inscribe<br />

circles <strong>in</strong> the figures bounded by the circle and the l<strong>in</strong>es AB and OP . Let<br />

R and S be the po<strong>in</strong>ts at which the two <strong>in</strong>circles to the curvil<strong>in</strong>ear triangles<br />

AOP and BOP are tangent to the diameter AB. Show that ∠RP S is<br />

<strong>in</strong>dependent of the position of O.<br />

Problem 67<br />

Let E be a po<strong>in</strong>t <strong>in</strong>side the triangle ABC such that ∠ABE = ∠ACE. Let F<br />

and G be the feet of the perpendiculars from E to the <strong>in</strong>ternal and external<br />

bisectors, respectively, of angle BAC. Prove that the l<strong>in</strong>e F G passes through<br />

the mid-po<strong>in</strong>t of BC.<br />

11


Problem 68<br />

Let A, B, C and D be po<strong>in</strong>ts on a circle with centre O and let P be the<br />

po<strong>in</strong>t of <strong>in</strong>tersection of AC and BD. Let U and V be the circumcentres<br />

of triangles AP B and CP D, respectively. Determ<strong>in</strong>e conditions on A, B,<br />

C and D that make O, U, P and V coll<strong>in</strong>ear and prove that, otherwise,<br />

quadrilateral OUP V is a parallelogram.<br />

Problem 69 [AMM]<br />

Let R and r be the circumradius and <strong>in</strong>radius, respectively of triangle ABC.<br />

(a) Show that ABC has a median whose length is at most 2R − r.<br />

(b) Show that ABC has an altitude whose length is at least 2R − r.<br />

Problem 70 [AMM]<br />

Let ABCD be a convex quadrilateral. Prove that if there is po<strong>in</strong>t P <strong>in</strong> the<br />

<strong>in</strong>terior of ABCD such that<br />

then ABCD is a square.<br />

Problem 71 [AMM]<br />

∠P AB = ∠P BC = ∠P CD = ∠P DA = 45 ◦<br />

Let M be any po<strong>in</strong>t <strong>in</strong> the <strong>in</strong>terior of triangle ABC and let D, E and F<br />

be po<strong>in</strong>ts on the perimeter such that AD, BE and CF are concurrent at<br />

M. Show that if triangles BMD, CME and AMF all have equal areas and<br />

equal perimeters then triangle ABC is equilateral.<br />

Problem 72<br />

The perpendiculars AD, BE, CF are produced to meet the circumscribed<br />

circle <strong>in</strong> X, Y , Z prove that<br />

Problem 73 [AMM]<br />

AX<br />

+ BY + CZ = 4<br />

AD BE CF<br />

Given an odd positive <strong>in</strong>teger n, let A 1 , A 2 ,...,A n be a regular polygon with<br />

circumcircle Γ. A circle O i with radius r is drawn externally tangent to Γ at<br />

A i for i = 1, 2, · · · , n. Let P be any po<strong>in</strong>t on Γ between A n and A 1 . A circle<br />

C (with any radius) is drawn externally tangent to Γ at P . Let t i be the<br />

length of the common external tangent between the circles C and O i . Prove<br />

that ∑ n<br />

i=1 (−1)i t i = 0.<br />

12


Problem 74 [INMO]<br />

The circumference of a circle is divided <strong>in</strong>to eight arcs by a convex quadrilateral<br />

ABCD, with four arcs ly<strong>in</strong>g <strong>in</strong>side the quadrilateral and the rema<strong>in</strong><strong>in</strong>g<br />

four ly<strong>in</strong>g outside it. The lengths of the arcs ly<strong>in</strong>g <strong>in</strong>side the quadrilateral<br />

are denoted by p, q, r, s <strong>in</strong> counter-clockwise direction start<strong>in</strong>g from some<br />

arc. Suppose p + r = q + s. Prove that ABCD is a cyclic quadrilateral.<br />

Problem 75 [INMO]<br />

In an acute-angled triangle ABC, po<strong>in</strong>ts D, E, F are located on the sides<br />

BC, CA, AB respectively such that<br />

CD<br />

= CA,<br />

AE = AB , BF = BC .<br />

CE CB AF AC BD BA<br />

Prove that AD, BE, CF are the altitudes of ABC.<br />

Problem 76<br />

In trapezoid ABCD, AB is parallel to CD and let E be the mid-po<strong>in</strong>t of<br />

BC. Suppose we can <strong>in</strong>scribe a circle <strong>in</strong> ABED and also <strong>in</strong> AECD. Then<br />

if we denote |AB| = a, |BC| = b, |CD| = c, |DA| = d prove that:<br />

Problem 77 [BMO]<br />

a + c = b 3 + d , 1 a + 1 c = 3 b .<br />

Let ABC be a triangle with AC > AB. The po<strong>in</strong>t X lies on the side BA<br />

extended through A and the po<strong>in</strong>t Y lies on the side CA <strong>in</strong> such a way that<br />

BX = CA and CY = BA. The l<strong>in</strong>e XY meets the perpendicular bisector<br />

of side BC at P . Show that<br />

Problem 78 [Loney]<br />

∠BP C + ∠BAC = 180 ◦<br />

If D, E, F are the po<strong>in</strong>ts of contact of the <strong>in</strong>scribed circle with the sides BC,<br />

CA, AB of a triangle, show that if the squares of AD, BE, CF are <strong>in</strong> arithmetic<br />

progression, then the sides of the triangle are <strong>in</strong> harmonic progression.<br />

13


Problem 79 [Loney]<br />

Through the angular po<strong>in</strong>ts of a triangle straight l<strong>in</strong>es mak<strong>in</strong>g the same angle<br />

α with the opposite sides are drawn. Prove that the area of the triangle<br />

formed by them is to the area of the orig<strong>in</strong>al triangle as 4 cos 2 α : 1.<br />

Problem 80 [Loney]<br />

If D, E, F be the feet of the perpendiculars from ABC on the opposite sides<br />

and ρ, ρ 1 , ρ 2 , ρ 3 be the radii of the circles <strong>in</strong>scribed <strong>in</strong> the triangles DEF ,<br />

AEF , BF D, CDE, prove that r 3 ρ = 2Rρ 1 ρ 2 ρ 3 .<br />

Problem 81 [Loney]<br />

A po<strong>in</strong>t O is situated on a circle of radius R and with centre O another<br />

circle of radius 3R is described. Inside the crescent-shaped area <strong>in</strong>tercepted<br />

2<br />

between these circles a circle of radius R is placed. Show that if the small<br />

8<br />

circle moves <strong>in</strong> contact with the orig<strong>in</strong>al circle of radius R, the length of arc<br />

described by its centre <strong>in</strong> mov<strong>in</strong>g from one extreme position to the other is<br />

7<br />

πR. 12<br />

Problem 82 [Crux]<br />

A Gergonne cevian is the l<strong>in</strong>e segment from a vertex of a triangle to the po<strong>in</strong>t<br />

of contact, on the opposite side, of the <strong>in</strong>circle. The Gergonne po<strong>in</strong>t is the<br />

po<strong>in</strong>t of concurrency of the Gergonne cevians.<br />

In an <strong>in</strong>teger triangle ABC, prove that the Gergonne po<strong>in</strong>t Γ bisects the<br />

Gergonne cevian AD if and only if b, c, |3a−b−c| form a triangle where the<br />

2<br />

measure of the angle between b and c is π.<br />

3<br />

Problem 83<br />

Prove that the l<strong>in</strong>e which divides the perimeter and the area of a triangle <strong>in</strong><br />

the same ratio passes through the centre of the <strong>in</strong>circle.<br />

Problem 84<br />

Let m a , m b , m c and w a , w b , w c denote, respectively, the lengths of the medians<br />

and angle bisectors of a triangle. Prove that<br />

√<br />

ma + √ m b + √ m c ≥ √ w a + √ w b + √ w c .<br />

14


Problem 85<br />

A quadrilateral has one vertex on each side of a square of side-length 1.<br />

Show that the lengths a, b, c and d of the sides of the quadrilateral satisfy<br />

the <strong>in</strong>equalities<br />

2 ≤ a 2 + b 2 + c 2 + d 2 ≤ 4.<br />

Problem 86 [Purdue Problem of the Week]<br />

Given a triangle ABC, f<strong>in</strong>d a triangle A 1 B 1 C 1 so that<br />

(1) A 1 ∈ BC, B 1 ∈ CA, C 1 ∈ AB<br />

(2) the centroids of triangles ABC and A 1 B 1 C 1 co<strong>in</strong>cide<br />

and subject to (1) and (2) triangle A 1 B 1 C 1 has m<strong>in</strong>imal area.<br />

Problem 87<br />

Prove that if the perpendiculars dropped from the po<strong>in</strong>ts A 1 , B 1 and C 1 on<br />

the sides BC, CA and AB of the triangle ABC, respectively, <strong>in</strong>tersect at the<br />

same po<strong>in</strong>t, then the perpendiculars dropped from the po<strong>in</strong>ts A, B and C on<br />

the l<strong>in</strong>es B 1 C 1 , C 1 A 1 and A 1 B 1 also <strong>in</strong>tersect at one po<strong>in</strong>t.<br />

Problem 88<br />

Drawn through the <strong>in</strong>tersection po<strong>in</strong>t M of medians of a triangle ABC is a<br />

straight l<strong>in</strong>e <strong>in</strong>tersect<strong>in</strong>g the sides AB and AC at po<strong>in</strong>ts K and L, respectively,<br />

and the extension of the side BC at a po<strong>in</strong>t P (C ly<strong>in</strong>g between P<br />

and B). Prove that<br />

Problem 89<br />

1<br />

= 1 + 1<br />

|MK| |ML| |MP |<br />

Prove that the area of the octagon formed by the l<strong>in</strong>es jo<strong>in</strong><strong>in</strong>g the vertices of<br />

a parallelogram to the midpo<strong>in</strong>ts of the opposite sides is 1/6 of the area of<br />

the parallelogram.<br />

Problem 90<br />

Prove that if the altitude of a triangle is √ 2 times the radius of the circumscribed<br />

circle, then the straight l<strong>in</strong>e jo<strong>in</strong><strong>in</strong>g the feet of the perpendiculars<br />

dropped from the foot of this altitude on the sides enclos<strong>in</strong>g it passes through<br />

the centre of the circumscribed circle.<br />

15


Problem 91<br />

Prove that the projections of the foot of the altitude of a triangle on the sides<br />

enclos<strong>in</strong>g this altitude and on the two other altitudes lie on one straight l<strong>in</strong>e.<br />

Problem 92<br />

Let a, b, c and d be the sides of an <strong>in</strong>scribed quadrilateral (a is opposite to<br />

c), h a , h b , h c and h d the distances from the centre of the circumscribed circle<br />

to the correspond<strong>in</strong>g sides. Prove that if the centre of the circle is <strong>in</strong>side the<br />

quadrilateral, then<br />

Problem 93<br />

ah c + ch a = bh d + dh b<br />

Prove that three l<strong>in</strong>es pass<strong>in</strong>g through the vertices of a triangle and bisect<strong>in</strong>g<br />

its perimeter <strong>in</strong>tersect at one po<strong>in</strong>t (called Nagell’s po<strong>in</strong>t). Let M denote<br />

the centre of mass of the triangle, I the centre of the <strong>in</strong>scribed circle, S the<br />

centre of the circle <strong>in</strong>scribed <strong>in</strong> the triangle with vertices at the midpo<strong>in</strong>ts of<br />

the sides of the given triangle. Prove that the po<strong>in</strong>ts N, M, I and S lie on<br />

a straight l<strong>in</strong>e and |MN| = 2|IM|, |IS| = |SN|.<br />

Problem 94 [Loney]<br />

If ∆ 0 be the area of the triangle formed by jo<strong>in</strong><strong>in</strong>g the po<strong>in</strong>ts of contact of<br />

the <strong>in</strong>scribed circle with the sides of the given triangle whose area is ∆ and<br />

∆ 1 , ∆ 2 and ∆ 3 the correspond<strong>in</strong>g areas for the escribed circles prove that<br />

Problem 95<br />

∆ 1 + ∆ 2 + ∆ 3 − ∆ 0 = 2∆<br />

Prove that the radius of the circle circumscribed about the triangle formed<br />

by the medians of an acute-angled triangle is greater than 5/6 of the radius<br />

of the circle circumscribed about the orig<strong>in</strong>al triangle.<br />

Problem 96<br />

Let K denote the <strong>in</strong>tersection po<strong>in</strong>t of the diagonals of a convex quadrilateral<br />

ABCD, L a po<strong>in</strong>t on the side AD, N a po<strong>in</strong>t on the side BC, M a po<strong>in</strong>t on<br />

the diagonal AC, KL and MN be<strong>in</strong>g parallel to AB, LM parallel to DC.<br />

Prove that KLMN is a parallelogram and its area is less than 8/27 of the<br />

area of the quadrilateral ABCD (Hattori’s Theorem).<br />

16


Problem 97<br />

Two triangles have a common side. Prove that the distance between the<br />

centres of the circles <strong>in</strong>scribed <strong>in</strong> them is less than the distance between<br />

their non-co<strong>in</strong>cident vertices (Zalgaller’s problem).<br />

Problem 98<br />

Prove that the sum of the distances from a po<strong>in</strong>t <strong>in</strong>side a triangle to its<br />

vertices is not less than 6r, where r is the radius of the <strong>in</strong>scribed circle.<br />

Problem 99<br />

Given a triangle. The triangle formed by the feet of its angle bisectors is<br />

isosceles. Is the given triangle isosceles?<br />

Problem 100<br />

Prove that the perpendicular bisectors of the l<strong>in</strong>e segments jo<strong>in</strong><strong>in</strong>g the <strong>in</strong>tersection<br />

po<strong>in</strong>ts of the altitudes to the centres of the circumscribed circles of<br />

the four triangles formed by four arbitrary straight l<strong>in</strong>es <strong>in</strong> the plane <strong>in</strong>tersect<br />

at one po<strong>in</strong>t (Herwey’s po<strong>in</strong>t).<br />

Problem 101 [Crux]<br />

Given triangle ABC with AB < AC. Let I be the <strong>in</strong>centre and M be the<br />

mid-po<strong>in</strong>t of BC. The l<strong>in</strong>e MI meets AB and AC at P and Q respectively.<br />

A tangent to the <strong>in</strong>circle meets sides AB and AC at D and E respectively.<br />

Prove that<br />

Problem 102 [Crux]<br />

AP<br />

+ AQ = P Q<br />

BD CE 2MI<br />

Let ABC be a triangle with ∠BAC = 60 ◦ . Let AP bisect ∠BAC and let<br />

BQ bisect ∠ABC, with P on BC and Q on AC. If AB + BP = AQ + QB,<br />

what are the angles of the triangle?<br />

Problem 103<br />

Prove that the sum of the squares of the distances from an arbitrary po<strong>in</strong>t <strong>in</strong><br />

the plane to the sides of a triangle takes on the least value for such a po<strong>in</strong>t<br />

<strong>in</strong>side the triangle whose distances to the correspond<strong>in</strong>g sides are proportional<br />

to these sides. Prove also that this po<strong>in</strong>t is the <strong>in</strong>tersection po<strong>in</strong>t of<br />

the symmedians of the given triangle (Lemo<strong>in</strong>e’s Po<strong>in</strong>t).<br />

17


Problem 104<br />

Given a triangle ABC. AA 1 , BB 1 and CC 1 are its altitudes. Prove that<br />

Euler’s l<strong>in</strong>es of the triangles AB 1 C 1 , A 1 BC 1 and A 1 B 1 C <strong>in</strong>tersect at a po<strong>in</strong>t<br />

P of the n<strong>in</strong>e-po<strong>in</strong>t circles such that one of the l<strong>in</strong>e segments P A 1 , P B 1 , P C 1<br />

is equal to sum of the other two (Thebault’s problem).<br />

Problem 105<br />

Let M be an arbitrary po<strong>in</strong>t <strong>in</strong> the plane and G, the centroid of triangle<br />

ABC. Prove that<br />

3|MG| 2 = |MA| 2 + |MB| 2 + |MC| 2 − 1 3 (|AB|2 + |BC| 2 + |CA| 2 )<br />

(Leibnitz’s Theorem)<br />

Problem 106<br />

Let ABC be a regular triangle with side a and M some po<strong>in</strong>t <strong>in</strong> the plane<br />

found at a distance d from the centre of the triangle ABC. Prove that the<br />

area of the triangle whose sides are equal to the l<strong>in</strong>e segments MA, MB and<br />

MC can be expressed by the formula<br />

Problem 107 [Todhunter]<br />

S = √ 3<br />

12 |a2 − 3d 2 |<br />

If Q be any po<strong>in</strong>t <strong>in</strong> the plane of a triangle and R 1 , R 2 , R 3 the radii of the<br />

circles about QBC, QCA, QAB prove that<br />

( a R 1<br />

+ b<br />

R 2<br />

+ c<br />

R 3<br />

)(− a R 1<br />

+ b<br />

R 2<br />

+ c<br />

R 3<br />

)( a R 1<br />

− b<br />

R 2<br />

+ c<br />

R 3<br />

)( a R 1<br />

+ b<br />

R 2<br />

− c<br />

R 3<br />

) = a2 b 2 c 2<br />

R 2 1 R2 2 R2 3<br />

Problem 108 [Mathematical Gazette]<br />

P QRS is a quadrilateral <strong>in</strong>scribed <strong>in</strong> a circle with centre O. E is the <strong>in</strong>tersection<br />

of the diagonals P R and QS. Let F be the<strong>in</strong>tersection of P Q and<br />

RS and G the <strong>in</strong>tersection of P S and QR. The circle on F G as diameter<br />

meets OE at X. The perpendicular bisectors of SX and P X meet at A and<br />

B, C, D are def<strong>in</strong>ed similarly by cyclic change of letters.<br />

(i) Prove that the tangents at P and Q and the l<strong>in</strong>e OB are concurrent.<br />

(ii) Prove that P Q, AC, SR, F G are concurrent at F .<br />

(iii)Prove that AD, BC, F G are concurrent.<br />

18


Problem 109 [AMM]<br />

Let X, Y and Z be three dist<strong>in</strong>ct po<strong>in</strong>ts <strong>in</strong> the <strong>in</strong>terior of an equilateral<br />

triangle ABC. Let α, β and γ be positive numbers add<strong>in</strong>g up to π with the<br />

3<br />

property that ∠XBA = ∠Y AB=α, ∠Y CB = ∠ZBC = β and ∠ZAC =<br />

∠XCA = γ. F<strong>in</strong>d the angles of triangle XY Z <strong>in</strong> terms of α, β and γ.<br />

Problem 110 [Todhunter]<br />

If O be the centre of the circle <strong>in</strong>scribed <strong>in</strong> a triangle ABC and r a , r b , r c the<br />

radii of the circles <strong>in</strong>scribed <strong>in</strong> the triangles OBC, OCA, OAB, show that<br />

Problem 111 [BMO]<br />

a<br />

r a<br />

+ b<br />

r b<br />

+ c<br />

r c<br />

= 2(cot( A) + cot( B ) + cot( C ))<br />

4 4 4<br />

Let P be an <strong>in</strong>ternal po<strong>in</strong>t of triangle ABC and let α, β, γ be def<strong>in</strong>ed by<br />

Prove that<br />

α = ∠BP C − ∠BAC<br />

β = ∠CP A − ∠CBA<br />

γ = ∠AP B − ∠ACB<br />

P A s<strong>in</strong>(∠BAC)<br />

s<strong>in</strong>(α)<br />

= P B s<strong>in</strong>(∠CBA)<br />

s<strong>in</strong>(β)<br />

= P C s<strong>in</strong>(∠ACB)<br />

s<strong>in</strong>(γ)<br />

Problem 112<br />

Let ABC be a triangle with <strong>in</strong>centre I and <strong>in</strong>radius r. Let D, E, F be the<br />

feet of the perpendiculars from I to the sides BC, CA and AB respectively.<br />

If r 1 , r 2 and r 3 are the radii of circles <strong>in</strong>scribed <strong>in</strong> the quadrilaterals AF IE,<br />

BDIF and CEID respectively, prove that<br />

Problem 113 [Loney]<br />

r 1<br />

r−r 1<br />

+ r 2<br />

r−r 2<br />

+ r 3<br />

r−r 3<br />

=<br />

r 1 r 2 r 3<br />

(r−r 1 )(r−r 2 )(r−r 3 )<br />

Given the product p of the s<strong>in</strong>es of the angles of a triangle and the product<br />

q of the cos<strong>in</strong>es, show that the tangents of the angles are the roots of the<br />

equation<br />

qx 3 − px 2 + (1 + q)x − p = 0<br />

19


Problem 114<br />

The altitude of a right triangle drawn to the hypotenuse is equal to h. Prove<br />

that the vertices of the acute angles of the triangle and the projections of<br />

the foot of the altitude on the legs all lie on the same circle. Determ<strong>in</strong>e the<br />

length of the chord cut by this circle on the l<strong>in</strong>e conta<strong>in</strong><strong>in</strong>g the altitude and<br />

the segments of the chord <strong>in</strong>to which it is divided by the hypotenuse.<br />

Problem 115<br />

Four villages are situated at the vertices of a square of side 2 Km. The<br />

villages are connected by roads so that each village is jo<strong>in</strong>ed to any other. Is<br />

it possible for the total length of the roads to be less than 5.5 Km?<br />

Problem 116<br />

Prove that if the lengths of the <strong>in</strong>ternal angle bisectors of a triangle are less<br />

than 1, then its area is less than √ 3<br />

3 .<br />

Problem 117<br />

Given a convex quadrilateral ABCD circumscribed about a circle of diameter<br />

1. Inside ABCD, there is a po<strong>in</strong>t M such that<br />

F<strong>in</strong>d the area of ABCD.<br />

Problem 118<br />

|MA| 2 + |MB| 2 + |MC| 2 + |MD| 2 = 2.<br />

The circle <strong>in</strong>scribed <strong>in</strong> a triangle ABC divides the median BM <strong>in</strong>to three<br />

equal parts. F<strong>in</strong>d the ratio |BC| : |CA| : |AB|.<br />

Problem 119<br />

Prove that if the centres of the squares constructed externally on the sides<br />

of a given triangle serve as the vertices of the triangle whose area is twice<br />

the area of the given triangle, then the centres of the squares constructed<br />

<strong>in</strong>ternally on the sides of the triangle lie on a straight l<strong>in</strong>e.<br />

Problem 120<br />

Prove that the median drawn to the largest side of a triangle forms with<br />

the sides enclos<strong>in</strong>g this median angles each of which is not less than half the<br />

smallest angle of the triangle.<br />

20


Problem 121<br />

Three squares BCDE, ACF G and BAHK are constructed externally on<br />

the sides BC, CA and AB of a triangle ABC. Let F CDQ and EBKP<br />

be parallelograms. Prove that the triangle AP Q is a right-angled isosceles<br />

triangle.<br />

Problem 122<br />

Three po<strong>in</strong>ts are given <strong>in</strong> a plane. Through these po<strong>in</strong>ts three l<strong>in</strong>es are drawn<br />

form<strong>in</strong>g a regular triangle. F<strong>in</strong>d the locus of centres of those triangles.<br />

Problem 123<br />

Drawn <strong>in</strong> an <strong>in</strong>scribed polygon are non-<strong>in</strong>tersect<strong>in</strong>g diagonals separat<strong>in</strong>g the<br />

polygon <strong>in</strong>to triangles. Prove that the sum of the radii of the circles <strong>in</strong>scribed<br />

<strong>in</strong> those triangles is <strong>in</strong>dependent of the way the diagonals are drawn.<br />

Problem 124<br />

A polygon is circumscribed about a circle. Let l be an arbitrary l<strong>in</strong>e touch<strong>in</strong>g<br />

the circle and co<strong>in</strong>cid<strong>in</strong>g with no side of the polygon. Prove that the ratio of<br />

the product of the distances from the verices of the polygon to the l<strong>in</strong>e l to<br />

the product of the distances from the po<strong>in</strong>ts of tangency of the sides of the<br />

polygon with the circle to l is <strong>in</strong>dependent of the position of the l<strong>in</strong>e l.<br />

Problem 125 [Loney]<br />

If 2φ 1 , 2φ 2 , 2φ 3 are the angles subtended by the circle escribed to the side a<br />

of a triangle at the centres of the <strong>in</strong>scribed circle and the other two escribed<br />

circles, prove that<br />

Problem 126<br />

s<strong>in</strong>(φ 1 ) s<strong>in</strong>(φ 2 ) s<strong>in</strong>(φ 3 ) = r2 1<br />

16R 2<br />

If from any po<strong>in</strong>t <strong>in</strong> the plane of a regular polygon perpendiculars are drawn<br />

on the sides, show that the sum of the squares of these perpendiculars is equal<br />

to the sum of the squares on the l<strong>in</strong>es jo<strong>in</strong><strong>in</strong>g the feet of the perpendiculars<br />

with the centre of the polygon.<br />

21


Problem 127 [Loney]<br />

The three medians of a triangle ABC make angles α, β, γ with each other.<br />

Prove that<br />

Problem 128 [Loney]<br />

cot α + cot β + cot γ + cot A + cot B + cot C = 0<br />

A railway curve, <strong>in</strong> the shape of a quadrant of a circle, has n telegraph posts<br />

at its ends and at equal distances along the curve. A man stationed at a<br />

po<strong>in</strong>t on one of the extreme radii produced sees the pth and qth posts from<br />

the end nearest him <strong>in</strong> a straight l<strong>in</strong>e. Show that the radius of the curve is<br />

a cos(p+q)φ<br />

, where φ = π , and a is the distance from the man to the<br />

2 s<strong>in</strong>(pφ) s<strong>in</strong>(qφ) 4(n−1)<br />

nearest end of the curve.<br />

Problem 129<br />

Let D be an arbitrary po<strong>in</strong>t on the side BC of a triangle ABC. Let E and F<br />

be po<strong>in</strong>ts on the sides AC and AB such that DE is parallel to AB and DF is<br />

parallel to AC. A circle pass<strong>in</strong>g through D, E and F <strong>in</strong>tersects for the second<br />

time BC, CA and AB at po<strong>in</strong>ts D 1 , E 1 and F 1 , respectively. Let M and N<br />

denote the <strong>in</strong>tersection po<strong>in</strong>ts of DE and F 1 D 1 , DF and D 1 E 1 , respectively.<br />

Prove that M and N lie on the symedian emanat<strong>in</strong>g from the vertex A. If<br />

D co<strong>in</strong>cides with the foot of the symedian, then the circle pass<strong>in</strong>g through<br />

D, E and F touches the side BC.(This circle is called Tucker’s Circle.)<br />

Problem 130<br />

Let ABCD be a cyclic quadrilateral. The diagonal AC is equal to a and<br />

forms angles α and β with the sides AB and AD, respectively. Prove that<br />

the magnitude of the area of the quadrilateral lies between a2 s<strong>in</strong>(α+β) s<strong>in</strong> β<br />

and<br />

2 s<strong>in</strong> α<br />

a 2 s<strong>in</strong>(α+β) s<strong>in</strong> α<br />

2 s<strong>in</strong> β<br />

Problem 131<br />

A triangle has sides of lengths a, b, c and respective altitudes of lengths h a ,h b ,<br />

h c . If a ≥ b ≥ c show that a + h a ≥ b + h b ≥ c + h c .<br />

22


Problem 132 [Crux]<br />

Given a right-angled triangle ABC with ∠BAC = 90 ◦ . Let I be the <strong>in</strong>centre<br />

and let D and E be the <strong>in</strong>tersections of BI and CI with AC and AB<br />

respectively. Prove that<br />

|BI| 2 +||ID| 2<br />

|IC| 2 +|IE| 2<br />

= |AB|2<br />

|AC| 2<br />

Problem 133 [Hobson]<br />

Straight l<strong>in</strong>es whose lengths are successively proportional to 1, 2, 3, · · · , n<br />

form a rectil<strong>in</strong>eal figure whose exterior angles are each equal to 2π n ; if a<br />

polygon be formed by jo<strong>in</strong><strong>in</strong>g the extremities of the first and last l<strong>in</strong>es, show<br />

that its area is<br />

Problem 134<br />

n(n+1)(2n+1)<br />

24<br />

cot( π n ) + n 16 cot( π n ) csc2 ( π n )<br />

An arc AB of a circle is divided <strong>in</strong>to three equal parts by the po<strong>in</strong>ts C and<br />

D (C is nearest to A). When rotated about the po<strong>in</strong>t A through an angle of<br />

π<br />

3 , the po<strong>in</strong>ts B, C and D go <strong>in</strong>to po<strong>in</strong>ts B 1, C 1 and D 1 . F is the po<strong>in</strong>t of<br />

<strong>in</strong>tersection of the straight l<strong>in</strong>es AB 1 and DC 1 ; E is a po<strong>in</strong>t on the bisector<br />

of the angle B 1 BA such that |BD| = |DE|. Prove that the triangle CEF is<br />

regular (F<strong>in</strong>lay’s theorem).<br />

Problem 135<br />

In a triangle ABC, a po<strong>in</strong>t D is taken on the side AC. Let O 1 be the centre<br />

of the circle touch<strong>in</strong>g the l<strong>in</strong>e segments AD, BD and the circle circumscribed<br />

about the triangle ABC and let O 2 be the centre of the circle touch<strong>in</strong>g the l<strong>in</strong>e<br />

segments CD, BD and the circumscribed circle. Prove that the l<strong>in</strong>e O 1 O 2<br />

passes through the centre O of the circle <strong>in</strong>scribed <strong>in</strong> the triangle ABC and<br />

|O 1 O| : |OO 2 | = tan 2 (φ/2), where φ = ∠BDA (Thebault’s theorem).<br />

Problem 136<br />

Prove the follow<strong>in</strong>g statement. If there is an n-gon <strong>in</strong>scribed <strong>in</strong> a circle α and<br />

circumscribed about another circle β, then there are <strong>in</strong>f<strong>in</strong>itely many n-gons<br />

<strong>in</strong>scribed <strong>in</strong> the circle α and circumscribed about the circle β and any po<strong>in</strong>t<br />

of the circle can be taken as one of the vertices of such an n-gon (Poncelet’s<br />

theorem).<br />

23


Problem 137 [Loney]<br />

A po<strong>in</strong>t is taken <strong>in</strong> the plane of a regular polygon of n sides at a distance c<br />

from the centre and on the l<strong>in</strong>e jo<strong>in</strong><strong>in</strong>g the centre to a vertex, and the radius<br />

of the <strong>in</strong>scribed circle is r. Show that the product of the distances of the<br />

po<strong>in</strong>t from the sides of the polygon is<br />

Problem 138 [Loney]<br />

cos 2 ( n 2 n−2 2 cos−1 r ) if c > r and<br />

c<br />

c n<br />

cosh 2 ( n 2 n−2 2 cosh−1 r) if c < r<br />

c<br />

c n<br />

An <strong>in</strong>f<strong>in</strong>ite straight l<strong>in</strong>e is divided by an <strong>in</strong>f<strong>in</strong>ite number of po<strong>in</strong>ts <strong>in</strong>to portions<br />

each of length a. Prove that the sum of the fourth powers of the<br />

reciprocals of the distances of a po<strong>in</strong>t O on the l<strong>in</strong>e from all the po<strong>in</strong>ts of<br />

division is<br />

Problem 139 [Loney]<br />

π 4<br />

(3 csc 4 πb<br />

πb<br />

− 2<br />

3a 4 csc2 )<br />

a<br />

a<br />

If ρ 1 , ρ 2 , · · · , ρ n be the distances of the vertices of a regular polygon of n sides<br />

from any po<strong>in</strong>t P <strong>in</strong> its plane, prove that<br />

1<br />

+ 1 + · · · + 1<br />

ρ 2 1 ρ 2 2<br />

ρ 2 n<br />

= n<br />

r 2n −a 2n<br />

r 2 −a 2 r 2n −2a n r n cos(nθ)+a 2n<br />

where a is the radius of the circumcircle of the polygon, r is the distance of<br />

P from its centre O and θ is the angle that OP makes with the radius to any<br />

angular po<strong>in</strong>t of the polygon.<br />

Problem 140<br />

Given an angle with vertex A and a circle <strong>in</strong>scribed <strong>in</strong> it. An arbitrary<br />

straight l<strong>in</strong>e touch<strong>in</strong>g the given circle <strong>in</strong>tersects the sides of the angle at<br />

po<strong>in</strong>ts B and C. Prove that the circle circumscribed about the triangle<br />

ABC touches the circle <strong>in</strong>scribed <strong>in</strong> the given angle.<br />

Problem 141<br />

Let ABCDEF be an <strong>in</strong>scribed hexagon <strong>in</strong> which |AB| = |CD| = |EF | = R,<br />

where R is the radius of the circumscribed circle, O its centre. Prove that<br />

the po<strong>in</strong>ts of pairwise <strong>in</strong>tersections of the circles circumscribed about the<br />

triangles BOC, DOE, F OA, dist<strong>in</strong>ct from O, serve as the vertices of an<br />

equilateral triangle with side R.<br />

24


Problem 142<br />

The diagonals of an <strong>in</strong>scribed quadrilateral are mutually perpendicular. Prove<br />

that the midpo<strong>in</strong>ts of its sides and the feet of the perpendiculars dropped<br />

from the po<strong>in</strong>t of <strong>in</strong>tersection of the diagonals on the sides lie on a circle.<br />

F<strong>in</strong>d the radius of that circle if the radius of the given circle is R and the<br />

distance from its centre to the po<strong>in</strong>t of <strong>in</strong>tersection of the diagonals of the<br />

quadrilateral is d.<br />

Problem 143<br />

Prove that if a quadrialateral is both <strong>in</strong>scribed <strong>in</strong> a circle and circumscribed<br />

about a circle of radius r, the distance between the centres of those circles<br />

be<strong>in</strong>g d, then the relationship<br />

is true.<br />

1<br />

+ 1 = 1<br />

(R+d) 2 (R−d) 2 r 2<br />

Problem 144<br />

Let ABCD be a convex quadrilateral. Consider four circles each of which<br />

touches three sides of this quadrilateral.<br />

(a) Prove that the centres of these circles lie on one circle.<br />

(b) Let r 1 , r 2 ,r 3 and r 4 denote the radii of these circles (r 1 does not touch<br />

the side DC, r 2 the side DA, r 3 the side AB and r 4 the side BC). Prove<br />

that<br />

|AB|<br />

r 1<br />

+ |CD|<br />

r 3<br />

= |BC|<br />

r 2<br />

+ |AD|<br />

r 4<br />

Problem 145<br />

The sides of a square is equal to a and the products of the distances from<br />

the opposite vertices to a l<strong>in</strong>e l are equal to each other. F<strong>in</strong>d the distance<br />

from the centre of the square to the l<strong>in</strong>e l if it is known that neither of the<br />

sides of the square is parallel to l.<br />

Problem 146<br />

F<strong>in</strong>d the angles of a triangle if the distance between the centre of the circumcircle<br />

and the <strong>in</strong>tersection po<strong>in</strong>t of the altitudes is one-half the length of<br />

the largest side and equals the smallest side.<br />

25


Problem 147<br />

Prove that for the perpendiculars dropped from the po<strong>in</strong>ts A 1 , B 1 and C 1 on<br />

the sides BC, CA and AB of a triangle ABC to <strong>in</strong>tersect at the same po<strong>in</strong>t,<br />

it is necessary and sufficient that<br />

Problem 148<br />

|A 1 B| 2 − |BC 1 | 2 + |C 1 A| 2 − |AB 1 | 2 + |B 1 C| 2 − |CA 1 | 2 = 0.<br />

Each of the sides of a convex quadrilateral is divided <strong>in</strong>to (2n + 1) equal<br />

parts. The division po<strong>in</strong>ts on the opposite sides are jo<strong>in</strong>ed correspond<strong>in</strong>gly.<br />

Prove that the area of the central quadrilateral amounts to 1/(2n + 1) 2 of<br />

the area of the entire quadrilateral.<br />

Problem 149<br />

A straight l<strong>in</strong>e <strong>in</strong>tersects the sides AB, BC and the extension of the side<br />

AC of a triangle ABC at po<strong>in</strong>ts D, E and F , respectively. Prove that<br />

the midpo<strong>in</strong>ts of the l<strong>in</strong>e segments DC, AE and BF lie on a straight l<strong>in</strong>e<br />

(Gaussian l<strong>in</strong>e).<br />

Problem 150<br />

Given two <strong>in</strong>tersect<strong>in</strong>g circles. F<strong>in</strong>d the locus of centres of rectangles with<br />

vertices ly<strong>in</strong>g on these circles.<br />

Problem 151<br />

An equilateral triangle is <strong>in</strong>scribed <strong>in</strong> a circle. F<strong>in</strong>d the locus of <strong>in</strong>tersection<br />

po<strong>in</strong>ts of the altitudes of all possible triangles <strong>in</strong>scribed <strong>in</strong> the circle if two<br />

sides of the triangles are parallel to those of the given one.<br />

Problem 152<br />

Given two circles touch<strong>in</strong>g each other <strong>in</strong>ternally at a po<strong>in</strong>t A. A tangent to<br />

the smaller circle <strong>in</strong>tersects the larger one at po<strong>in</strong>ts B and C. F<strong>in</strong>d the locus<br />

of centres of circles <strong>in</strong>scribed <strong>in</strong> triangles ABC.<br />

Problem 153 [Loney]<br />

Two circles, the sum of whose radii is a, are placed <strong>in</strong> the same plane with<br />

their centres at a distance 2a and an endless str<strong>in</strong>g is fully stretched so as<br />

partly to surround the circles and to cross between them. Show that the<br />

length of the str<strong>in</strong>g is ( 4π + 2√ 3)a.<br />

3<br />

26


Problem 154 [Loney]<br />

If p, q, r are the perpendiculars from the vertices of a triangle upon any<br />

straight l<strong>in</strong>e meet<strong>in</strong>g the sides externally <strong>in</strong> D, E, F , prove that<br />

a 2 (p − q)(p − r) + b 2 (q − r)(q − p) + c 2 (r − p)(r − q) = 4∆ 2 .<br />

Problem 155 [Loney]<br />

A regular polygon is <strong>in</strong>scribed <strong>in</strong> a circle; show that the arithmetic mean of<br />

the squares of the distances of its corners from any po<strong>in</strong>t (not necessarily <strong>in</strong><br />

its plane) is equal to the arithmetic mean of the sum of the squares of the<br />

longest and shortest distances of the po<strong>in</strong>t from the circle.<br />

Problem 156<br />

In the cyclic quadrilateral ABCD, the diagonal AC bisects the angle DAB.<br />

The side AD is extended beyond D to a po<strong>in</strong>t E. Show that CE = CA if<br />

and only if DE = AB.<br />

Problem 157 [BMO]<br />

Let G be a convex quadrilateral. Show that there is a po<strong>in</strong>t X <strong>in</strong> the plane<br />

of G with the property that every straight l<strong>in</strong>e through X divides G <strong>in</strong>to two<br />

regions of equal area if and only if G is a parallelogram.<br />

Problem 158<br />

Given a triangle ABC and a po<strong>in</strong>t M. A straight l<strong>in</strong>e pass<strong>in</strong>g through the<br />

po<strong>in</strong>t M <strong>in</strong>tersects the l<strong>in</strong>es AB, BC and CA at po<strong>in</strong>ts C 1 , A 1 and B 1 ,<br />

respectively. The l<strong>in</strong>es AM, BM and CM <strong>in</strong>tersect the circle circumscribed<br />

about the triangle ABC at po<strong>in</strong>ts A 2 , B 2 and C 2 , respectively. Prove that<br />

the l<strong>in</strong>es A 1 A 2 , B 1 B 2 and C 1 C 2 <strong>in</strong>tersect at a po<strong>in</strong>t situated on the circle<br />

circumscribed about the triangle ABC.<br />

Problem 159 [AMM]<br />

Let P be a po<strong>in</strong>t <strong>in</strong> the <strong>in</strong>terior of triangle ABC and let r 1 , r 2 , r 3 denote<br />

the distances from P to the sides of the triangle with lengths a 1 , a 2 , a 3 ,<br />

respectively. Let R be the circumradius of ABC and let 0 < a < 1 be a real<br />

number. Let b = 2a/(1 − a). Prove that<br />

r a 1 + r a 2 + r a 3 ≤ 1<br />

(2R) a (a b 1 + a b 2 + a b 3) 1−a<br />

27


Problem 160 [AMM]<br />

Let K be the circumcentre and G the centroid of a triangle with side lengths<br />

a, b, c and area ∆.<br />

(a) Show that the distance d from K to G satisfies<br />

12∆d = a 2 b 2 c 2 − (b 2 + c 2 − a 2 )(c 2 + a 2 − b 2 )(a 2 + b 2 − c 2 )<br />

(b) Show that d(< abc , = abc , > abc ) when the triangle is respectively<br />

12∆ 12∆ 12∆<br />

(acute, right-angled,obtuse).<br />

Problem 161 [AMM]<br />

Let ABC be an acute-triangle and let P be a po<strong>in</strong>t <strong>in</strong> its <strong>in</strong>terior. Denote by<br />

a, b, c the lengths of the triangle’s sides, by d a , d b , d c the distances from P to<br />

the triangle’s sides, and by R a , R b , R c the distances from P to the vertices<br />

A, B, C respectively. Show that<br />

d 2 a +d 2 b +d2 c ≥ R 2 as<strong>in</strong> 2 (A/2)+R 2 b s<strong>in</strong>2 (B/2)+R 2 cs<strong>in</strong> 2 (C/2) ≥ (d a +d b +d c ) 2 /3<br />

Problem 162 [Loney]<br />

A 1 A 2 · · · A n is a regular polygon of n sides which is <strong>in</strong>scribed <strong>in</strong> a circle, whose<br />

radius is a and whose centre is O; prove that the product of the distances of<br />

its angular po<strong>in</strong>ts from a straight l<strong>in</strong>e at right angles to OA and at a distance<br />

b(> a) from the centre is<br />

Problem 163 [Loney]<br />

b n [cos n ( 1 2 s<strong>in</strong>−1 a b ) − s<strong>in</strong>n ( 1 2 s<strong>in</strong>−1 a b )]2<br />

The radii of an <strong>in</strong>f<strong>in</strong>ite series of concentric circles are a, a, a · · · . From a po<strong>in</strong>t<br />

2 3<br />

at a distance c(> a) from their common centre a tangent is drawn to each<br />

circle. Prove that<br />

s<strong>in</strong>(θ 1 ) s<strong>in</strong>(θ 2 ) s<strong>in</strong>(θ 3 ) · · · = √ c πa<br />

s<strong>in</strong><br />

πa c<br />

where θ 1 , θ 2 , θ 3 , · · · are the angles that the tangents subtend at the common<br />

centre.<br />

28


Problem 164 [Crux]<br />

Construct equilateral triangles A ′ BC, B ′ CA, C ′ AB exterior to triangle ABC<br />

and take po<strong>in</strong>ts P , Q, R on AA ′ , BB ′ , CC ′ , respectively, such that<br />

Prove that ∆P QR is equilateral.<br />

Problem 165 [Crux]<br />

AP<br />

+ BQ + CR = 1.<br />

AA ′ BB ′ CC ′<br />

Given a triangle ABC, we take variable po<strong>in</strong>ts P on segment AB and Q on<br />

segment AC. CP meets BQ <strong>in</strong> T . Where should P and Q be located so that<br />

area of ∆P QT is maximized?<br />

Problem 166 [Crux]<br />

Let ABC be a triangle and A 1 , B 1 , C 1 the common po<strong>in</strong>ts of the <strong>in</strong>scribed<br />

circle with the sides BC, CA, AB, respectively. We denote the length of the<br />

arc B 1 C 1 (not conta<strong>in</strong><strong>in</strong>g A 1 ) of the <strong>in</strong>circle by S a , and similarly def<strong>in</strong>e S b<br />

and S c . Prove that<br />

Problem 167 [AMM]<br />

a<br />

S a<br />

+ b<br />

S b<br />

+ c<br />

S c<br />

≥ 9√ 3<br />

π<br />

A cevian of a triangle is a l<strong>in</strong>e segment that jo<strong>in</strong>s a vertex to the l<strong>in</strong>e conta<strong>in</strong><strong>in</strong>g<br />

the opposite side. An equicevian po<strong>in</strong>t of a triangle ABC is a po<strong>in</strong>t<br />

P (not necessarily <strong>in</strong>side the triangle) such that the cevians on the l<strong>in</strong>es AP ,<br />

BP and CP have equal length. Let SBC be an equilateral triangle and let<br />

A be chosen <strong>in</strong> the <strong>in</strong>terior of SBC on the altitude dropped from S.<br />

(a) Show that ABC has two equicevian po<strong>in</strong>ts.<br />

(b) Show that the common length of the cevians through either of the<br />

equicevian po<strong>in</strong>ts is constant, <strong>in</strong>dependent of the choice of A.<br />

(c) Show that the equicevian po<strong>in</strong>ts divide the cevian through A <strong>in</strong> a<br />

constant ratio, <strong>in</strong>dependent of the choice of A.<br />

(d) F<strong>in</strong>d the locus of the equicevian po<strong>in</strong>ts as A varies.<br />

(e) Let S ′ be the reflection of S <strong>in</strong> the l<strong>in</strong>e BC. Show that (a), (b) and (c)<br />

hold if A moves on any ellipse with S and S ′ as its foci. F<strong>in</strong>d the locus of<br />

the equicevian po<strong>in</strong>ts as A varies on the ellipse.<br />

29


Problem 168 [Crux]<br />

Let ABCD be a trapezoid with AD parallel to BC. M, N, P , Q, O are the<br />

midpo<strong>in</strong>ts of AB, CD, AC, BD, MN, respectively. Circles m, n, p, q all<br />

pass through O and are tangent to AB at M, to CD at N, to AC at P , and<br />

to BD at Q, respectively. Prove that the centres of m, n, p, q are coll<strong>in</strong>ear.<br />

Problem 169 [AMM]<br />

Let ABC be an equilateral triangle <strong>in</strong>scribed <strong>in</strong> a circle with radius 1 unit.<br />

Suppose P is a po<strong>in</strong>t <strong>in</strong>side the triangle. Prove that |P A||P B||P C| ≤ 32.<br />

27<br />

Generalize the result to a regular polygon of n sides. (Erdos)<br />

Problem 170<br />

Given two circles. F<strong>in</strong>d the locus of po<strong>in</strong>ts M such that the ratio of the<br />

lengths of the tangents drawn from M to the given circles is a constant k.<br />

Problem 171<br />

In a quadrilateral ABCD, P is the <strong>in</strong>tersection po<strong>in</strong>t of BC and AD, Q<br />

that of CA and BD and R that of AB and CD. Prove that the <strong>in</strong>tersection<br />

po<strong>in</strong>ts of BC and QR, CA and RP , AB and P Q are coll<strong>in</strong>ear.<br />

Problem 172<br />

Given two squares whose sides are respectively parallel. Determ<strong>in</strong>e the locus<br />

of po<strong>in</strong>ts M such that for any po<strong>in</strong>t P of the first square there is a po<strong>in</strong>t Q of<br />

the second one such that the triangle MP Q is equilateral. Let the side of the<br />

first square be a and that of the second square be b. For what relationship<br />

between a and b is the desired locus non-empty?<br />

Problem 173 [AMM]<br />

Let C 1 C 2 . . . C n be a regular n-gon and let C n+1 = C 1 . Let O be the <strong>in</strong>scribed<br />

circle. For 1 ≤ k ≤ n, let T k be the po<strong>in</strong>t at which O is tangent to C k C k+1 .<br />

Let X be a po<strong>in</strong>t on the open arc (T n−1 T n ) and let Y be a po<strong>in</strong>t other than<br />

X on O. For 1 ≤ i ≤ n, let B i be the second po<strong>in</strong>t at which the l<strong>in</strong>e XC i<br />

meets O and let p i = |XB i ||XC i |. Let M i be the mid-po<strong>in</strong>t of chord T i T i+1<br />

and let N i be the second po<strong>in</strong>t, other than Y , at which Y M i meets O. Let<br />

∑<br />

q i = |Y M i ||Y N i |. Prove that n ∑<br />

q i = ( n p i ) − p n .<br />

i=1<br />

i=1<br />

30


Problem 174 [AMM]<br />

Let ABC be an acute triangle, with semi-perimeter p and with <strong>in</strong>scribed and<br />

circumscribed circles of radius r and R, respectively.<br />

(a) Show that ABC has a median of length at most p/ √ 3.<br />

(b) Show that ABC has a median of length at most R + r.<br />

(c) Show that ABC has an altitude of length at least R + r.<br />

Problem 175 [RMO, India]<br />

Let ABC be an acute-angled triangle and CD be the altitude through C. If<br />

AB = 8 and CD = 6 f<strong>in</strong>d the distance between the mid-po<strong>in</strong>ts of AD and<br />

BC.<br />

Problem 176 [RMO, India]<br />

Let ABCD be a rectangle with AB = a and BC = b. Suppose r 1 is the radius<br />

of the circle pass<strong>in</strong>g through A and B and touch<strong>in</strong>g CD; and similarly r 2 is<br />

the radius of the circle pass<strong>in</strong>g through B and C and touch<strong>in</strong>g AD. Show<br />

that<br />

Problem 177 [INMO]<br />

r 1 + r 2 ≥ 5 (a + b)<br />

8<br />

Two circles C 1 and C 2 <strong>in</strong>tersect at two dist<strong>in</strong>ct po<strong>in</strong>ts P and Q <strong>in</strong> a plane. Let<br />

a l<strong>in</strong>e pass<strong>in</strong>g through P meet the circles C 1 and C 2 <strong>in</strong> A and B respectively.<br />

Let Y be the mid-po<strong>in</strong>t of AB and QY meet the circles C 1 and C 2 <strong>in</strong> X and<br />

Z respectively. Show that Y is also the mid-po<strong>in</strong>t of XZ.<br />

Problem 178 [INMO]<br />

In a triangle ABC angle A is twice angle B. Show that a 2 = b(b + c).<br />

Problem 179 [INMO]<br />

The diagonals AC and BD of a cyclic quadrilateral ABCD <strong>in</strong>tersect at P .<br />

Let O be the circumcentre of triangle AP B and H be the orthocentre of<br />

triangle CP D. Show that the po<strong>in</strong>ts H, P , O are coll<strong>in</strong>ear.<br />

31


Problem 180 [INMO]<br />

Let ABC be a triangle <strong>in</strong> a plane Σ. F<strong>in</strong>d the set of all po<strong>in</strong>ts P (dist<strong>in</strong>ct<br />

from A, B, C) <strong>in</strong> the plane Σ such that the circumcircles of triangles ABP ,<br />

BCP and CAP have the same radii.<br />

Problem 181 [INMO]<br />

Let ABC be a triangle right-angled at A and S be its circumcircle. Let S 1 be<br />

the circle touch<strong>in</strong>g the l<strong>in</strong>es AB and AC and the circle S <strong>in</strong>ternally. Further<br />

let S 2 be the circle touch<strong>in</strong>g the l<strong>in</strong>es AB and AC and the circle S externally.<br />

If r 1 and r 2 be the radii of the circles S 1 and S 2 respectively, show that<br />

Problem 182 [INMO]<br />

r 1 .r 2 = 4Area(ABC)<br />

Show that there exists a convex hexagon <strong>in</strong> the plane such that<br />

(a) all its <strong>in</strong>terior angles are equal.<br />

(b) all its sides are 1,2,3,4,5,6 <strong>in</strong> some order.<br />

Problem 183 [INMO]<br />

Let G be the centroid of a triangle ABC <strong>in</strong> which the angle C is obtuse and<br />

AD and CF be the medians from A and C respectively onto the sides BC<br />

and AB. If the four po<strong>in</strong>ts B, D, G and F are concyclic, show that<br />

AC<br />

> √ 2<br />

BC<br />

If further P is a po<strong>in</strong>t on the l<strong>in</strong>e BG extended such that AGCP is a parallelogram,<br />

show that the triangle ABC and GAP are similar.<br />

Problem 184 [INMO]<br />

A circle passes through a vertex C of a rectangle ABCD and touches its<br />

sides AB and AD at M and N respectively. If the distance from C to the<br />

l<strong>in</strong>e segment MN is equal to 5 units, f<strong>in</strong>d the area of the rectangle ABCD.<br />

32


Problem 185 [RMO, India]<br />

In a quadrilateral ABCD, it is given that AB is parallel to CD and the<br />

diagonals AC and BD are perpendicular to each other. Show that<br />

(a) AD.BC ≥ AB.CD<br />

(b) AD + BC ≥ AB + CD<br />

Problem 186 [RMO, India]<br />

In the triangle ABC, the <strong>in</strong>circle touches the sides BC, CA and AB respectively<br />

at D, E and F . If the radius of the <strong>in</strong>circle is 4 units and if BD, CE<br />

and AF are consecutive <strong>in</strong>tegers , f<strong>in</strong>d the sides of the triangle ABC.<br />

Problem 187 [RMO, India]<br />

Let ABCD be a square and M, N po<strong>in</strong>ts on sides AB, BC, respectively,<br />

such that ∠MDN = 45 ◦ . If R is the midpo<strong>in</strong>t of MN show that RP = RQ<br />

where P , Q are the po<strong>in</strong>ts of <strong>in</strong>tersection of AC with the l<strong>in</strong>es MD and ND.<br />

Problem 188 [RMO, India]<br />

Let AC and BD be two chords of a circle with centre O such that they<br />

<strong>in</strong>tersect at right angles <strong>in</strong>side the circle at the po<strong>in</strong>t M. Suppose K and L<br />

are the mid-po<strong>in</strong>ts of the chord AB and CD respectively. Prove that OKML<br />

is a parallelogram.<br />

Problem 189 [AMM]<br />

Let P be a convex n-gon <strong>in</strong>scribed <strong>in</strong> a circle O and let ∆ be a triangulation<br />

of P without new vertices. Compute the sum of the squares of distances from<br />

the centre O to the <strong>in</strong>centres of the triangles of ∆ and show that this sum is<br />

<strong>in</strong>dependent of ∆.<br />

Problem 190 [AMM]<br />

Let T 1 and T 2 be triangles such that for i ∈ 1, 2, triangle T i has circumradius<br />

R i , <strong>in</strong>radius r i and side lengths a i , b i and c i . Show that<br />

8R 1 R 2 + 4r 1 r 2 ≥ a 1 a 2 + b 1 b 2 + c 1 c 2 ≥ 36r 1 r 2<br />

and determ<strong>in</strong>e when equality holds.<br />

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Problem 191 [AMM]<br />

Let ABC be an acute triangle. T the mid-po<strong>in</strong>t of arc BC of the circle<br />

circumscrib<strong>in</strong>g ABC. Let G and K be the projections of A and T respectively<br />

on BC, let H and L be the projections of B and C on AT and let E be the<br />

mid-po<strong>in</strong>t of AB. Prove that:<br />

(a) KH||AC, GL||BT , GH||T C, LK||AB.<br />

(b)G, H, K and L are concyclic.<br />

(c) The centre of the circle through G, H and K lies on the Euler circle of<br />

ABC.<br />

Problem 192 [AMM]<br />

A trapezoid ABRS with AB||RS is <strong>in</strong>scribed <strong>in</strong> a non-circular ellipse E with<br />

axes of symmetry a and b. The po<strong>in</strong>ts A and B are reflected through a to<br />

po<strong>in</strong>ts P and Q on E.<br />

(a) Show that P , Q, R and S are concyclic.<br />

(b) Show that if the l<strong>in</strong>e P Q <strong>in</strong>tersects the l<strong>in</strong>e RS at T , then the angle<br />

bisector of ∠P T R is parallel to a.<br />

Problem 193 [RMO, India]<br />

ABCD is a cyclic quadrilateral with AC ⊥ BD; AC meets BD at E. Prove<br />

that EA 2 + EB 2 + EC 2 + ED 2 = 4R 2 .<br />

Problem 194 [RMO, India]<br />

ABCD is a cyclic quadrilateral; x, y, z are the distances of A from the l<strong>in</strong>es<br />

BD, BC, CD respectively. Prove that<br />

Problem 195 [RMO, India]<br />

BD<br />

x<br />

= BC<br />

y<br />

+ CD z<br />

ABCD is a quadrilateral and P , Q are mid-po<strong>in</strong>ts of CD, AB respectively.<br />

Let AP , DQ meet at X and BP , CQ meet at Y . Prove that<br />

area(ADX)+area(BCY ) = area(P XQY ).<br />

34


Problem 196 [RMO, India]<br />

The cyclic octagon ABCDEF GH has sides a, a, a, a, b, b, b, b respectively.<br />

F<strong>in</strong>d the radius of the circle that circumscribes ABCDEF GH <strong>in</strong> terms of a<br />

and b.<br />

Problem 197 [AMM]<br />

Prove that <strong>in</strong> an acute triangle with angles A, B and C<br />

(1−cos A)(1−cos B)(1−cos C)<br />

cos A cos B cos C<br />

≥<br />

8(tan A+tan B+tan C) 3<br />

27(tan A+tan B)(tan C+tan A)(tan B+tan C)<br />

Problem 198 [Mathscope, Vietnam]<br />

In a triangle ABC, denote by l a , l b , l c the <strong>in</strong>ternal angle bisectors, m a , m b ,<br />

m c the medians and h a , h b , h c the altitudes to the sides a, b, c of the triangle.<br />

Prove that<br />

m a<br />

l b +h b<br />

+<br />

m b<br />

h c+l c<br />

+<br />

Problem 199 [Mathscope, Vietnam]<br />

mc<br />

l a+h a<br />

≥ 3 2<br />

Let AM, BN, CP be the medians of triangle ABC. Prove that if the radius<br />

of the <strong>in</strong>circles of triangles BCN, CAP and ABM are equal <strong>in</strong> length, then<br />

ABC is an equilateral triangle.<br />

Problem 200 [Mathscope, Vietnam]<br />

Given a triangle with <strong>in</strong>centre I, let l be a variable l<strong>in</strong>e pass<strong>in</strong>g through I.<br />

Let l <strong>in</strong>tersect the ray CB, sides AC, AB at M, N, P respectively. Prove<br />

that the value of<br />

AB<br />

+<br />

P A.P B<br />

is <strong>in</strong>dependent of the choice of l.<br />

AC −<br />

NA.NC<br />

Problem 201 [Mathscope, Vietnam]<br />

BC<br />

MB.MC<br />

Let I be the <strong>in</strong>centre of triangle ABC and let m a , m b , m c be the lengths of<br />

the medians from vertices A, B and C, respectively. Prove that<br />

IA 2<br />

m 2 a<br />

+ IB2<br />

m 2 b<br />

+ IC2<br />

m 2 c<br />

≤ 3 4<br />

35


Problem 202 [Mathscope, Vietnam]<br />

Let R and r be the circumradius and <strong>in</strong>radius of triangle ABC; the <strong>in</strong>circle<br />

touches the sides of the triangle at three po<strong>in</strong>ts which form a triangle of<br />

perimeter p. Suppose that q is the perimeter of triangle ABC. Prove that<br />

Problem 203 [AMM]<br />

r<br />

R ≤ p q ≤ 1 2<br />

Let a, b and c be the lengths of the sides of a triangle and let R and r be the<br />

circumradius and <strong>in</strong>radius of that triangle, respectively. Show that<br />

R<br />

2r<br />

Problem 204 [AMM]<br />

≥ exp(<br />

(a−b)2<br />

2c 2<br />

+ (b−c)2<br />

2a 2<br />

+ (c−a)2<br />

2b 2 )<br />

Consider an acute triangle with sides of lengths a, b and c and with an<br />

<strong>in</strong>radius of r and circumradius of R. Show that<br />

√<br />

r<br />

≤ 2(2a 2 −(b−c) 2 )(2b 2 −(c−a) 2 )(2c 2 −(a−b) 2 )<br />

.<br />

R (a+b)(b+c)(c+a)<br />

Problem 205 [AMM]<br />

Let a, b and c be the lengths of the sides of a triangle, and let R and r denote<br />

the circumradius and <strong>in</strong>radius of the triangle. Show that<br />

Problem 206 [AMM]<br />

R<br />

≥ ( 4a 2 4b 2 4c 2<br />

) 2<br />

2r 4a 2 −(b−c) 2 4b 2 −(c−a) 2 4c 2 −(a−b) 2<br />

Let ABC be a triangle with sides a, b and c all different, and correspond<strong>in</strong>g<br />

angles α, β and γ. Show that<br />

(a) (a + b) cot(β + γ 2 ) + (b + c) cot(γ + α 2 ) + (c + a) cot(α + β 2 ) = 0.<br />

(b) (a − b) tan(β + γ ) + (b − c) tan(γ + α) + (c − a) tan(α + β ) = 4(R + r).<br />

2 2 2<br />

Problem 207 [AMM]<br />

Let r, R and s be the radii of the <strong>in</strong>circle, circumcircle and semi-perimeter<br />

of a triangle. Prove that<br />

√<br />

3√<br />

r r2 s ≤<br />

2 +4Rr<br />

≤ s 3 3<br />

Problem 208 [AMM]<br />

Let a, b and c be the lengths of the sides of a nondegenerate triangle, let<br />

p = (1/2)(a + b + c), and let r and R be the <strong>in</strong>radius and circumradius of the<br />

triangle, respectively. Show that<br />

a<br />

2 ( 4r−R<br />

R ) ≤ √ (p − b)(p − c) ≤ a 2 .<br />

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