Proposition 20. Theorem. 318. If in any triangle a parallel be drawn to the base, all lines from the vertex will divide the base and its parallel proportionally. Hyp. Let ABC be a A, B'C' || to BC, and AD, AE two lines intersecting B'C' at D'E'. AS AD'E' and ADE; and the AS AE'C' and AEC. (311) = Q.E.D. 319. COR. If BD DE EC, we shall have B'D' = D'E' E'C'. Therefore, if the lines from the vertex divide the base into equal parts, they will also divide the parallel into equal parts. Proposition 21. Theorem. 320. If two polygons are composed of the same number of triangles similar each to each, and similarly placed, the polygons are similar. Hyp. Let the As ABC, ACD, ADE of the polygon ABCDE be similar respectively to the As A'B'C', A'C'D', A'D'E' of the polygon A'B'C'D'E', and similarly placed. To prove the polygons are similar. Proof. Since the homologous /s of similars are equal, (307) A Α E E' B In like manner, CDE = C'D'E'; etc. .. the polygons are mutually equiangular. Since similar As have their homologous sides propor tional, ... the homologous sides of the polygons are proportional. ... the polygons are similar, being mutually equiangular and having their homologous sides Proposition 22. Theorem. 321. Conversely, two similar polygons may be divided into the same number of triangles, similar each to each, and similarly placed. Hyp. Let ABCDE, A'B'C'D'E' be two similar polygons divided into As by the diagonals AC, AD, A'C', A'D' drawn from the homologous s A and A'. To prove As ABC, ACD, ADE sim ilar respectively to As A'B'C', A'C'D', A'D'E'. Proof. Since the polygons are similar, A E B E' Α In the same way it may be shown that As ADE, A'D'E' are similar. Q.E.D. COR. The homologous diagonals of two similar polygons are proportional to the homologous sides. Proposition 23. Theorem. 322. The perimeters of two similar polygons are to each other as any two homologous sides. Hyp. Let ABCDE, A'B'C'D'E' be two similar polygons; denote their perimeters by P and P'. To prove P: P' = AB : A'B'. Proof. Since the polygons are similar, E B Ε ́ 1. From the ends of a side of a triangle any two straight lines are drawn to meet the other sides in P, Q; also from the same ends two lines parallel to the former are drawn to meet the sides produced in P', Q': show that PQ is parallel to P'Q'. Let ABC be the ; BP, CQ the lines. AC: AP = AQ AB, etc. 2. ABC is a triangle, AD any line drawn from A to a point D in BC; a line is drawn from B bisecting AD in E and cutting AC in F: prove that BF is to BE as 2 CF is to AC. Draw EG || to AC meeting BC in G. DE EA, .. AC = 2 EG, etc. 3. If in Prop. 20, AD = 24, AD' = 20, AE = 16, BD = 8, and DE 6, find B'D', D'E', and AE'. = NUMERICAL RELATIONS BETWEEN THE DIFFERENT Proposition 24. Theorem. 323. In a right triangle, if a perpendicular be drawn from the right angle to the hypotenuse: (1) The two triangles on each side of it are similar to the whole triangle and to each other. (2) The perpendicular is a mean proportional between the segments of the hypotenuse. (3) Each side about the right angle is a mean proportional between the hypotenuse and the adjacent segment. Hyp. Let ABC be a rt. A, and AD the from the rt. A to the hypotenuse BC. (1) To prove the As DAB, DAC, and ABC similar. Proof. In the rt. As DAB, .. the As DAB, DAC are similar to each other, as |