Plane Geometry - Bruce E. Shapiro

254 SECTION 45. EUCLIDEAN CONSTRUCTIONSFigure 45.3: Construction 45.2. A copy of ∠EDF on ray −−→ DE is constructedsuch that ∠EDF ∼ = ∠BAC.5. Draw an arc of c 3 = C(B, r).6. Circles c 2 and c 3 intersect at two points: one of these points is A; callthe other point Q.7. Draw line n = ←→ P Q. Then n ‖ l.Construction 45.8 (Partition a Segment) Given a line segment AB anda positive integer n, divide the segment into n segments of equal length.1. Pick any point Q that is not on the line ←→ AB.2. Construct ray −→ AQ.3. Pick any point Q 1 ∈ −→ AQ.4. Define r = AQ 1 .5. Define points Q 2 , Q 3 , ..., Q n ∈ −→ AQ such that Q i Q i+1 = r.6. Construct line m n = ←−→ Q n B.7. Construct lines m 1 , m 2 , ..., m n−1 through points Q 1 , Q 2 , ..., Q n−1 thatare parallel to to m n .8. Define the points P 1 = m 1 ∩ −→ AB, P2 = m 2 ∩ −→ AB,..., Pn−1 = m n−1 ∩−→AB.9. Then the segments AP 1 , P 1 P 2 ,...,P n−2 P n−1 , P n−1 B are the desiredsegments of equal length.Construction 45.9 (Tangent to a Circle) Given a circle γ = C(O, r)« CC BY-NC-ND 3.0. Revised: 18 Nov 2012