Читать книгу Geometry for Students and Parents - Sergey D Skudaev - Страница 10

A perpendicular line drawn from a midpoint of a trianglés side is called a perpendicular bisector. See Figure 18.

Figure 18. Midpoints are: D, H, F; ED _|_AC; JH_|_AB; GF_|_BC; AD = DC; AH = HB; BF=FC;

Perpendicular bisectors of the sides of a triangle are concurrent. The point, at which the perpendicular bisectors intersect, is called the circumcenter. A circumcenter of a triangle is equidistant from the vertices of the triangle and consequently it is a center of a circumscribed circle. See figure 19.

Figure 19. O is a center of the circumscribed circle.

A point that lies on a perpendicular bisector of a triangle side is equidistant from the endpoints of the side. Converse also is true. See figure 20.

Figure 20. A perpendicular bisector BD.

If DB is a perpendicular bisector of the side AC then AE = EC and AF=FC.

In any triangle, the orthocenter, centroid and circumcenter lie on the same line which is called the Euler line named after a Swiss mathematician and physicist Leonhard Euler. The distance from orthocenter O to centroid I is twice as long as the distance from centroid I to circumcenter C. See figure 21.

Figure 21. Euler Line. (CIO)

IO=2IC. GF, JH, ED are perpendicular bisectors. Point C is circumcenter.

AI, BI, KI are medians. Point I is centroid. AO, BO, CO are perpendiculars. Point O is orthocenter.