HMMT 二月 2002 · 几何 · 第 5 题

HMMT February 2002 — Geometry — Problem 5

Consider a square of side length 1. Draw four lines that each connect a midpoint of a side with a corner not on that side, such that each midpoint and each corner is touched by only one line. Find the area of the region completely bounded by these lines.

解析

Consider a square of side length 1. Draw four lines that each connect a midpoint of a side with a corner not on that side, such that each midpoint and each corner is touched by only one line. Find the area of the region completely bounded by these lines. 1 Solution: In unit square ABCD , denote by E, F, G, H the respective midpoints of sides AB, BC, CD, DA . Let I be the intersection of AF and DE , let J be the intersection of BG and AF , let K be the intersection of CH and BG , and let L be the intersection of DE and 1 CH . We want to find the area of square IJKL . The area of ABF is , which is equal to 4 √ 1 5 1 1 √ AF · BJ = BJ , so BJ = . Using similar triangles, GK = JF = BJ . Thus the 2 4 2 5 √ 5 1 1 1 1 1 √ √ √ length of a side of IJKL is JK = − − = , and the area of IJKL is . 2 2 5 5 5 5