HMMT 二月 2002 · 几何 · 第 7 题
HMMT February 2002 — Geometry — Problem 7
题目详情
英文原题
- Equilateral triangle ABC of side length 2 is drawn. Three squares external to the triangle, ABDE , BCF G , and CAHI , are drawn. What is the area of the smallest trianglethat contains these squares?
解析
英文解析
- Equilateral triangle ABC of side length 2 is drawn. Three squares external to the triangle, ABDE , BCF G , and CAHI , are drawn. What is the area of the smallest trianglethat contains these squares?
Solution: The equilateral triangle with sides lying on lines DG , EH , and F I has minimalarea. (The only other reasonable candidate is the triangle with sides along DE, F G, HI , buta quick sketch shows that it is larger.) Let J , K , and L be the vertices of this triangle closest
° ° °
to D , H , and F , respectively. Clearly, KI = F L = 2. Triangle F CI is a 30 − 30 − 120
√
triangle, so we can calculate the length of F I as 2 3, making the side length of 4 JKL
√ √
4 + 2 3, and its area 12 + 7 3 .