HMMT 二月 2019 · 几何 · 第 2 题

HMMT February 2019 — Geometry — Problem 2

In rectangle ABCD , points E and F lie on sides AB and CD respectively such that both AF and CE are perpendicular to diagonal BD . Given that BF and DE separate ABCD into three polygons with equal area, and that EF = 1, find the length of BD .

解析

In rectangle ABCD , points E and F lie on sides AB and CD respectively such that both AF and CE are perpendicular to diagonal BD . Given that BF and DE separate ABCD into three polygons with equal area, and that EF = 1, find the length of BD . Proposed by: Yuan Yao √ Answer: 3 1 Observe that AECF is a parallelogram. The equal area condition gives that BE = DF = AB . 3 √ EX BE 1 2 2 Let CE ∩ BD = X , then = = , so that BX = EX · CX = 3 EX ⇒ BX = 3 EX ⇒ CX CD 3 3 3 ◦ ∠ EBX = 30 . Now, CE = 2 BE = CF , so CEF is an equilateral triangle and CD = CF = . 2 2 √ 2 3 √ Hence, BD = · = 3. 2 3