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