HMMT 二月 2002 · 几何 · 第 7 题

HMMT February 2002 — Geometry — Problem 7

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 triangle that contains these squares?

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 triangle that contains these squares? Solution: The equilateral triangle with sides lying on lines DG , EH , and F I has minimal area. (The only other reasonable candidate is the triangle with sides along DE, F G, HI , but a 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 .