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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.

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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

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M and N (Fig. 3) are not similar, even if the homologous

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between homologous sides is the ratio of similitude of the figures.

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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.

207. Theorem.

PROPOSITION XV

Two triangles are similar if their

homologous sides are proportional.

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PROPOSITION XVI

208. Theorem. Two triangles are similar if an angle of one equals an angle of the other and the including sides are proportional.

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