HMMT 二月 2002 · 几何 · 第 8 题
HMMT February 2002 — Geometry — Problem 8
题目详情
英文原题
- 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 thatcontains these squares?
解析
英文解析
- 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 trianglethat contains these squares?
Solution: The equilateral triangle with sides lying on lines DE , F G , and HI has minimalarea. 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 .