HMMT 二月 2001 · 几何 · 第 9 题

英文解析

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 hasmarea m and perimeter n . Compute the value of if AF : AD = 1 : 4.
   n 2
   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 findthat DO : Y O = 7 : 1. Therefore, the rhombus is composed of four triangles, whose sides
   √ √
   are in the ratio 1 : 7 : 5 2. The perimeter of the rhombus is 20 2 N , and the area is 14 N .2
   The required ratio is thus .7
   400