HMMT 二月 2001 · 几何 · 第 9 题

HMMT February 2001 — Geometry — Problem 9

Parallelogram AECF is inscribed in square ABCD . It is reflected across diagonal ′ ′ AC to form another parallelogram AE CF . The region common to both parallelograms has m area m and perimeter n . Compute the value of if AF : AD = 1 : 4. 2 n

解析

Parallelogram AECF is inscribed in square ABCD . It is reflected across diagonal ′ ′ AC to form another parallelogram AE CF . The region common to both parallelograms has m area m and perimeter n . Compute the value of if AF : AD = 1 : 4. 2 n Solution: By symmetry, the region is a rhombus, AXCY, centered at the center of the square, O . Consider isoceles right triangle ACD . By the technique of mass points, we find that DO : Y O = 7 : 1. Therefore, the rhombus is composed of four triangles, whose sides √ √ 2 are in the ratio 1 : 7 : 5 2. The perimeter of the rhombus is 20 2 N , and the area is 14 N . 7 The required ratio is thus . 400