HMMT 二月 2000 · 几何 · 第 7 题
HMMT February 2000 — Geometry — Problem 7
题目详情
- Let AB C b e a triangle ins rib ed in the ellipse + = 1. If its en troid is the origin 4 9 (0,0), nd its area. p 15
解析
- Let's "resize" the o ordinates to b e x = x; y = . This k eeps the origin at (0,0), but 3 turns our ellipse in to a ir le of radius 2. Th us no w, the triangle is equilateral, and w e A h G 1 h 2 F C B b 2 E D p an see it no w has area 3 3. On e w e expand ba k, w e an see w e are just m ultiplying p 9 3 3 the area b y and so the answ er is . 2 2 ab sin A