215. The bisector of any angle of a triangle divides the opposite sides into segments proportional to the adjacent sides. C G AL B Let ABC be the triangle, and GB the bisector of the angle ABC. To prove AG: GC AB : BC. = Draw CD parallel to GB, and produce AB to D. hence (by 93) the triangle BCD is isosceles, and EXERCISES. 1. If a line divides one side of a triangle into segments that are proportional to the adjacent sides, it bisects the opposite angle. 2. The bisector of an exterior angle of a triangle divides the opposite side externally into segments E B F 5. Bisectors of an interior and exterior angle at the vertex of a triangle divide the opposite side harmonically. 216. DEFINITIONS. Two polygons are called Similar when they are mutually equiangular (122) and have their homologous sides proportional (122). That is, the polygons ABCDE and A'B'C'D'E' are similar if: LA=ZA', LB = LB', / C = C', etc., and AB BC CD = A'B' B'C' C'D" etc. 217. In two similar polygons, the ratio of any two homologous sides is called the Ratio of Similitude of the polygons. PROPOSITION XIII. THEOREM. 218. Triangles which are mutually equiangular are similar. Let ABC and A'B'C' be two equi angular triangles. To prove that ABC and A'B'C' are similar triangles. B A D E CB Lay off on AB a distance equal to A'B' and on AC make AE equal to A'C'. C' The triangles ADE and A'B'C' are (by 86) equal, having the included A equal to Z A', and the sides AD and AE equal to A'B' and A'C", by construction. Therefore ADE = ≤ B', but by hypothesis, BB', hence ADE = B, therefore (by 62) DE is parallel to BC. If DE is parallel to BC, we have (by 214) or substituting for AD its equal A'B', and for AE, A'C", AB: A'B' = AC: A'C'. Similarly, it can be shown, by laying off on BA a distance equal to B'A', and on BC a distance equal to B'C', that BA : B'A' = BC : B'C'. Q.E.D. 219. COR. 1. Two triangles are similar when two angles of the one are equal respectively to two angles of the other. 220. COR. 2. A triangle is similar to any triangle cut off by a line parallel to one of its sides. 221. SCHOLIUM. In similar triangles the homologous sides lie opposite the equal angles. PROPOSITION XIV. THEOREM. 222. Two triangles are similar when their homologous sides are proportional. Take ADA'B' and AE = A'C', and join DE. therefore (by converse of 213, Ex. 1) the line DE is parallel to BC, and the angles ADE and B having their sides parallel and similarly directed are (by 63) equal; likewise, AED = C. Hence the triangles ADE and ABC are mutually equiangular and (by 218) are similar; that is, These last two proportions agree term for term except the last in each, and they must be equal or B'C' = DE. Hence the triangles ADE and A'B'C' are mutually equilateral and therefore equal. But the triangle ADE has been proved similar to ABC. Q.E.D. 223. SCHOLIUM. Two polygons are similar when they are mutually equiangular and have their homologous sides proportional. But in the case of triangles we learn, from Propositions XIII. and XIV., that either of these conditions involves the other. This, however, is not necessarily the case with polygons of more than three sides; for even with quadrilaterals, the angles can be changed without altering the sides, or the proportionality of the sides can be changed without altering the angles. EXERCISES. 1. Two right triangles are similar when they have an acute angle of one equal to an acute angle of the other. 2. Two triangles are similar when they have an angle of one equal to an angle of the other, and the sides including these angles proportional. 3. Two triangles are similar when the sides of one are parallel respectively to the sides of the other. 4. Two triangles are similar when the sides of one are perpendicular respectively to the sides of the other. SUGGESTION. See 63. 5. The homologous altitudes of two similar triangles have the same ratio as any two homologous sides. 6. If in any triangle a parallel be drawn to the base, all lines from the vertex will divide the base and its parallel proportionally. SUGGESTION. See 218. A B' C'D' A B 7. Two parallelograms are similar when they have an angle equal and the including sides proportional. 8. Two rectangles are similar when they have two adjacent sides proportional. 9. If two triangles stand upon the same base, and not between the same parallels, the figure formed by joining the middle points of their sides is a parallelogram. 10. If from any two diametrically opposite points on the circumference of a circle perpendiculars be drawn to a straight line outside the circle, the sum of these perpendiculars is constant, |