PROPOSITION XXIV. THEOREM 362 If two polygons are composed of the same number of triangles, similar each to each, and similarly placed, the polygons are similar. HYPOTHESIS. The ABC, ACD, and ADE are similar respectively to the ▲ A'B'C', A'C'D', and A'D'E', and similarly placed. CONCLUSION. The polygons ABCDE and A'B'C'D'E' are similar. In like manner it may be shown that the other corresponding angles of the polygons are equal. That is, the polygons are mutually equiangular. That is, the homologous sides of the polygons are proportional. .. the polygons are similar. "Similar polygons are polygons which have their homologous angles equal and their homologous sides proportional." § 349 Q. E. D. PROPOSITION XXV. THEOREM 363 If two polygons are similar, they are composed of the same number of triangles, similar each to each, and similarly placed. HYPOTHESIS. The polygons ABCDE and A'B'C'D'E' are similar. CONCLUSION. The AABC, ACD, and ADE are similar respectively to the A A'B'C', A'C'D', and A'D'E'. Subtracting, and BCA = ▲ B'C'A'. ACD=ZA'C'D'. Again, BC: B'C' = AC : A'C', and BC: B'C' = CD : C'D'. .. AC : A'C' = CD : C'D'. .. the ▲ ACD and A'C'D' are similar. Likewise the A ADE and A'D'E' are similar. Q. E. D. 364 COROLLARY. Two similar polygons are equal, if any $ 349 Ax. 4 $ 349 Ax. 11 $ 354 two homologous lines are equal. PROPOSITION XXVI. THEOREM 365 If in a right triangle an altitude is drawn to the hypotenuse: 1 The two triangles thus formed are similar to the whole triangle and to each other. 2 The altitude is the mean proportional between the segments of the hypotenuse. 3 Each leg of the right triangle is the mean proportional between the hypotenuse and the adjacent segment. HYPOTHESIS. In the rt. ▲ BAC, AD is to BC, the hypotenuse. PROOF The rt. A BDA and BAC have the Z B common; ... they are similar. The rt. A ADC and BAC have the C common; .. they are similar. The ABDA and ADC are similar, for each is similar to the ▲ BAC. § 353 § 353 Proved 2 AD is the mean proportional between BD and DC. PROOF In the similar ▲ BDA and ADC, BD: AD = AD : DC. § 349 3 BA is the mean proportional between BC and BD, and CA is the mean proportional between CB and CD. 366 COROLLARY 1. The squares of the legs of a right triangle are proportional to the adjacent segments of the hypote nuse. PROOF From the proportions in the third part of the theorem, § 328 Q. E. D. 367 COROLLARY 2. The altitude upon the hypotenuse of a right triangle is equal to the product of the legs divided by the hypotenuse. 368 COROLLARY 3. The perpendicular drawn from any point in the circumference of a circle to the diameter is the mean proportional between the segments of the diameter. PROPOSITION XXVII. THEOREM 369 The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs. HYPOTHESIS. BC is the hypotenuse of the rt. ▲ BAC. 370 COROLLARY 1. The square of either leg of a right triangle is equal to the difference of the squares of the hypotenuse and the other leg. 371 COROLLARY 2. The side and the diagonal of a square are incommensurable. d2 = s2 + s2 = 2s2. . d = s√2. B 372 DEFINITION. The projection of a line AB upon a line MN is that part of MN included between the feet of the perpendiculars drawn from A and B to MN. Thus, CD is the projection of AB upon MN. M -N C D |