PROP. LXXX. THEOR. quadrilateral figure inscribed in a circle, Let ABCD be any quadrilateral fig. inscrib. in a 0 ; draw the diagonals AC, BD: then shall AC. BD = AB.CD + AD.BC. PROP. LXXXI. THEOR. Equiangular triangles have their bases in the same ratio as their altitudes, or perpendiculars upon the bases from the opposite and equal angles. Let the As ABC, DEF have the Ls A, B, C of the one A, respectively equal to the Ls D, E, F of the other; draw AG I BC and DH I EF; then shall DH : AG :: EF: BC; or DH : EF :: AG : BC. PROP. LXXXII. THEOR. Triangles having equal bases are to each other as their altitudes. Let ACB, DFE be 2 As, having base AB = base DE. Draw CH | AB, and FI IDF, then shall AACB : ADFE :: CH : FI. Р Join SAP make JPB = CH, JAP AQ ADFE, - having equal Prop. 35. .. A ACB: ADFE:: CH: FI. Cor. 1. PROP. LXXXIII. THEOR. 19. 6 Eu. Equiangular triangles are to each other as the squares of their corresponding sides. Let As ABC, DEF have the Ls A, B, C, = Ls D, E, F respectively; then shall A ABC : A DEF :: BCP : EF. On BC and EF draw the squares BK, EM, and the diams. CI, FL; Prop. 38. SAABC AG PROP. LXXXIV. THEOR. Similar rectilinear figures are to each other as the squares of their like or corresponding sides. Let ABCDE, FGHIK be any two similar figures, having the Zs A, B, C, &c., of the one figure respectively equal to the Ls F, G, H, &c., of the other, and the sides about the equal Zs proportional; also let AB and FG be corresponding sides; then shall fig. ABCDE : fig. FGHIK :: AB? : FG%. |