...quan• tity ^p, and reducing, we have ACD : PQR :: CD2 : QR2. But, CD2:QR2::DA2:RP2::AC2:PQ2. c Hence, two similar triangles are to each other as the squares of any two homologous sides. Cor. 2. If ACD and PQR are similar, ACD and EFD are also similar; consequently the second couplets...

...similar triangles, homologous altitudes are to each other as any pair of homologous sides. XXVI. Theorem. The areas of two similar triangles are to each other as the squares of two homologous sides or altitudes. abc. ABC : abc = HYPOTH. A ABC ~ To IJE PROVED. AB2 : a&2. PROOF....

...triangles ABC and DEF be similar, the angle A being equal to the angle D, B to E, and C to F : then the triangles are to each other as the squares of any two homologous sides. Because the angles A and D are equal, we have (P. XXTV.), ABC : DEF : : ABxAC : DE x DF ; and,...

...hence, CD AB C'D' A'B' Substituting in (1), we have ABC = A'B'C' A'B AB AB A'B' AB2 335. SCHOLIUM. Two similar triangles are to each other as the squares of any two homologous lines. PROPOSITION VIII. THEOREM. 336. Two triangles having an angle of one equal to an angle of the other,...

...equalities, AABC_ABxAC AADE ADxAE §370 q. ED COMPARISON OF POLYGONS. PROPOSITION VIII. THEOREM. 375. The areas of two similar triangles are to each other as the squares of any two homologous sides. Let the two triangles be ACS and A'CB'. A ACB AJ? Draw the perpendiculars CO and C'O'. A ACB...

...equal ^57^ . D .A. r> O A ABC BC BC BC' ' ' ' A A'B'C' ~ B'C' X B'C' - g^c'' • Q•KD• 378. Cor. The areas of two similar triangles are to each other as the squares of any two homologous lines. EXERCISES. 1. ABC is a triangle. AE and BF, intersecting in G, are drawn to bisect the sides BC, AC...

...in the above equality for -=^-, its equal T^^-, A ABC BC BC _ BC' A A'B'C' " B'C' 378. Cor. The arms of two similar triangles are to each other as the squares of any two homologous lines. EXERCISES. 1. ABC is a triangle. AE and BF, intersecting in G, are drawn to bisect the sides BC, AC...

...A ABC BC BC EC ' 1 ' A A'B'C 7 " B'C' B'C' ~~ B/C"' ' ' '• 7 " '' X _ 378. Cor. The areas of (wo similar triangles are to each other as the squares of any two homologous lines. EXERCISES. 1. ABC is a triangle. AE and BF, intersecting in G, are drawn to bisect the sides BC, AC...

...PLANE GEOMETRY. — BOOK IV. COMPARISON OF POLYGONS. PROPOSITION VIII. THEOREM. 375. The areas of tivo similar triangles are to each other as the squares of any two homologous sides. A 0 B A! O' B' Let the two triangles be ACB and A'GB'. A ACB Al? To prove A A'C'B' Draw the...

...formed by a straight line parallel to the base, and distant 5 feet from the vertex ? 2. Prove that the areas of two similar triangles are to each other as the squares of the radii of their inscribed circles. 3. Prove that the area of a circumscribed polygon is equal...