HMMT 二月 2008 · 几何 · 第 2 题

HMMT February 2008 — Geometry — Problem 2

[ 3 ] Let ABC be an equilateral triangle. Let Ω be its incircle (circle inscribed in the triangle) and let ω be a circle tangent externally to Ω as well as to sides AB and AC . Determine the ratio of the radius of Ω to the radius of ω . ◦

解析

[ 3 ] Let ABC be an equilateral triangle. Let Ω be its incircle (circle inscribed in the triangle) and let ω be a circle tangent externally to Ω as well as to sides AB and AC . Determine the ratio of the radius of Ω to the radius of ω . Answer: 3 Label the diagram as shown below, where Ω and ω also denote the center of the corresponding 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 scale factor of 3. A ω ′ ′ B C N Ω B C M ◦