HMMT 二月 2002 · 几何 · 第 8 题

HMMT February 2002 — Geometry — Problem 8

Equilateral triangle ABC of side length 2 is drawn. Three squares containing 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 containing 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 DE , F G , and HI has minimal area. Let J , K , and L be the vertices of this triangle closest to D , H , and F , respectively. ◦ ◦ ◦ Clearly, DE = 2. Denote by M the intersection of CI and BD . Using the 30 − 30 − 120 √ √ triangle BCM , we get BM = 2 / 3, and thus M D = 2 − 2 / 3. By symmetry, M J bisects √ ◦ ◦ ◦ angle DM I , from which we see that 4 JM D is a 30 − 60 − 90 . We then get JD = 2 3 − 2, √ √ making the side length of JKL 4 3 − 2, and its area 13 3 − 12 .