HMMT 二月 2008 · 几何 · 第 2 题

英文解析

2. [ 3 ] Let ABC be an equilateral triangle. Let Ω be its incircle (circle inscribed in the triangle) and let ωbe a circle tangent externally to Ω as well as to sides AB and AC . Determine the ratio of the radiusof Ω to the radius of ω .
   Answer: 3 Label the diagram as shown below, where Ω and ω also denote the center of thecorresponding circles. Note that AM is a median and Ω is the centroid of the equilateral triangle.
   So AM = 3 M Ω. Since M Ω = N Ω, it follows that AM/AN = 3, and triangle ABC is the image of
   ′ ′
   triangle AB C after a scaling by a factor of 3, and so the two incircles must also be related by a scalefactor of 3.
   ′ ′ωA
   B C
   B CΩN
   °M