Plane Geometry - Bruce E. Shapiro
Plane Geometry - Bruce E. Shapiro Plane Geometry - Bruce E. Shapiro
182 SECTION 36. TRIANGLE CENTERSFigure 36.1: The intersection of the perpendicular bisectors of the sides ofa triangle is called the orthocenter of the triangle.Thus ∠ARO = ∠CRO. But these angles are supplements, hence RO ⊥AC.Thus all three side bisectors pass through the point O. We have alreadyshown that O is equidistant from A, B, and C.Definition 36.3 Altitude. Let l be a line that is constructed perpendicularto any side of a triangle that is concurrent with the vertex V oppositethat side. Then l is said to be the line containing the altitude of thetriangle. The altitude is the distance from V to l.Theorem 36.4 The lines containing the altitudes of any triangle intersectat a point called the orthocenter.Proof. The construction is illustrated in figure 36.2. Construct line l ‖ ABthrough C; line m ‖ BC through A; and line n ‖ AC through B.Define the intersections l ∩ n = A ′ , l ∩ n = B ′ , and m ∩ n = C ′ as shownin figure 36.2.By construction, □ABA ′ C is a parallelogram, so that BA ′ = AC.By construction □C ′ BCA is a parallelogram, so that BC ′ = AC.Hence BA ′ = BC ′ , so that B is the midpoint of A ′ C ′ .Since the altitude through B is perpendicular to AC, and AC ‖ ←−→ A ′ C ′ , weconclude that the altitude through B is the perpendicular bisector of A ′ C ′ .« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 36. TRIANGLE CENTERS 183By a similar argument, the altitude through C is the perpendicular bisectorof A ′ B ′ , and the altitude through A is the perpendicular bisector of B ′ C ′ .By theorem 36.2, these three bisectors of the sides of triangle △A ′ B ′ C ′must meet at a common point O.Thus the altitudes of the original triangle △ABC meet at a common pointO.Figure 36.2: The orthocenter is the intersection of the three altitudes of atriangle.Figure 36.3: The three medians of a triangle all meet at the centroid of thetriangle.Definition 36.5 A median of a triangle is a line segment whose endpointsare one vertex and the midpoint of the side opposite that vertex.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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SECTION 36. TRIANGLE CENTERS 183By a similar argument, the altitude through C is the perpendicular bisectorof A ′ B ′ , and the altitude through A is the perpendicular bisector of B ′ C ′ .By theorem 36.2, these three bisectors of the sides of triangle △A ′ B ′ C ′must meet at a common point O.Thus the altitudes of the original triangle △ABC meet at a common pointO.Figure 36.2: The orthocenter is the intersection of the three altitudes of atriangle.Figure 36.3: The three medians of a triangle all meet at the centroid of thetriangle.Definition 36.5 A median of a triangle is a line segment whose endpointsare one vertex and the midpoint of the side opposite that vertex.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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