HMMT 二月 2001 · 几何 · 第 8 题

HMMT February 2001 — Geometry — Problem 8

Point D is drawn on side BC of equilateral triangle ABC , and AD is extended past D to E such that angles EAC and EBC are equal. If BE = 5 and CE = 12, determine the length of AE .

解析

Point D is drawn on side BC of equilateral triangle ABC , and AD is extended past D to E such that angles EAC and EBC are equal. If BE = 5 and CE = 12, determine the length of AE . Solution: By construction, ABEC is a cyclic quadrilateral. Ptolemy’s theorem says that for cyclic quadrilaterals, the sum of the products of the lengths of the opposite sides equals the product of the lengths of the diagonals. This yields ( BC )( AE ) = ( BA )( CE ) + ( BE )( AC ). Since ABC is equilateral, BC = AC = AB , so dividing out by this common value we get AE = CE + BE = 17 .