PROPOSITION XIII 202. Theorem. The bisector of an exterior angle of a triangle divides the opposite side externally into segments proportional to the adjacent sides. 203. Sch. If a line is divided so that the ratio between its internal segments equals the ratio between its external segments, the line is said to be divided harmonically. The bisector of an angle of a triangle and the bisector of the exterior angle at the same vertex divide the opposite side of the triangle harmonically, because the ratios between the segments of this side, both internal and external, equal the ratio between the adjacent sides. SIMILAR POLYGONS 204. Definition. Two polygons are similar when they are mutually equiangular and have their homologous sides proportional. B Fig. 1. M A 23 54 Fig. 2. B AB BC CD and etc. 12 34' M and N (Fig. 2) are N с not similar, even if A=1, B= 2, etc. AE ED unless which 15 E D is evidently not true. M and N (Fig. 3) are not similar, even if the homologous sides are propor B tional, because the homologous 2x A 1 205. Similar N 20 5 4 shape. The ratio between homologous sides is the ratio of similitude of the figures. 2 Y C X Y 3 22 z or U FIG. 3. PROPOSITION XIV 206. Theorem. Two triangles are similar if they are mutually equiangular. 170. In a triangle ABC, AB = 6, BC = 10, and CA = 12. Find the segments of CA, formed by a bisector of the angle B. |