PROPOSITION XVII. THEOREM. 296. The homologous altitudes of two similar triangles have the same ratio as any two homologous sides. In the two similar triangles ABC and A'B'C', let the altitudes be BO and B'O'. (being homologous of the similar ▲ A B C and A' B' C'). $278 297. COR. 1. The homologous altitudes of similar triangles have the same ratio as their homologous bases. (the homologous sides of similar ▲ are proportional). And in the similar A BOA and B'O' A', $278 § 296 Ax. 1 298. COR. 2. The homologous altitudes of similar triangles have the same ratio as their perimeters. Denote the perimeter of the first by P, and that of the second by P'. (the perimeters of two similar polygons have the same ratio as any two Ex. 1. If any two straight lines be cut by parallel lines, show that the corresponding segments are proportional. 2. If the four sides of any quadrilateral be bisected, show that the lines joining the points of bisection will form a parallelogram. 3. Two circles intersect; the line AHKB joining their centres A, B, meets them in H, K. On A B is described an equilateral triangle ABC, whose sides BC, A C, intersect the circles in F, E. FE produced meets BA produced in P. Show that as PA is to PK so is CF to CE, and so also is PH to PB. 299. In any triangle the product of two sides is equal to the product of the segments of the third side formed by the bisector of the opposite angle together with the square of the bisector. B Di E Let BAC of the ▲ A B C be bisected by the straight line A D. We are to prove BAX ACBD X D C + A D2. Describe the C A B C about the ▲ A BC; produce A D to meet the circumference in E, and draw E C. Then in the AA BD and A EC, (two are similar when two of the one are equal respectively to two $ 290 (the segments of two chords in a which intersect each other are reciprocally proportional). Substitute in the above equality BDX DC for ED × A D, then BAX ACB D X D C + A D2. Q. E. D. PROPOSITION XIX. THEOREM. 300. In any triangle the product of two sides is equal to the product of the diameter of the circumscribed circle by the perpendicular let fall upon the third side from the vertex of the opposite angle. Let ABC be a triangle, and AD the perpendicular from A to BC. Describe the circumference ABC about the ▲ ABC. Draw the diameter A E, and draw E C. $ 281 (each being measured by the arc A C). ..A ABD and A E C are similar, (two rt. A having an acute of the one equal to an acute of the other are PROPOSITION XX. THEOREM. 301. The product of the two diagonals of a quadrilateral inscribed in a circle is equal to the sum of the products of its opposite sides. B E C We are to prove Construct Let ABCD be any quadrilateral inscribed in a circle, AC and BD its diagonals. = BDXAC ABXCD+ADX BC. LABEL DBC, |