HMMT 二月 1998 · 几何 · 第 6 题
HMMT February 1998 — Geometry — Problem 6
题目详情
英文原题
Question Six . [5 points]
A circle is inscribed in an equilateraltriangle of side length 1. Tangents tothe circle are drawn that cut offequilateral triangles at each corner.
Circles are inscribed in each of theseequilateral triangles. If this process isrepeated infinitely many times, what isthe sum of the areas of all the circles?
解析
英文解析
- The central circle has area π = , and each of the three small triangles are copies of
2 312
( )
1 π 12
the entire figure dilated by . Therefore, the total area is given by K = + 3 · K ⇐⇒
3 12 3
2 π π
K = ⇐⇒ K = .
3 12 8