HMMT 二月 2003 · 几何 · 第 9 题

HMMT February 2003 — Geometry — Problem 9

In triangle ABC , ABC = 50 and ACB = 70 . Let D be the midpoint of side BC . A circle is tangent to BC at B and is also tangent to segment AD ; this circle instersects AB again at P . Another circle is tangent to BC at C and is also tangent 6 to segment AD ; this circle intersects AC again at Q . Find AP Q (in degrees). 6 6

解析

In triangle ABC , ABC = 50 and ACB = 70 . Let D be the midpoint of side BC . A circle is tangent to BC at B and is also tangent to segment AD ; this circle instersects AB again at P . Another circle is tangent to BC at C and is also tangent 6 to segment AD ; this circle intersects AC again at Q . Find AP Q (in degrees). Solution: 70 Suppose the circles are tangent to AD at E, F , respectively; then, by equal tangents, DE = DB = DC = DF ⇒ E = F (as shown). So, by the Power of a Point Theorem, 2 2 AP · AB = AE = AF = AQ · AC ⇒ AP/AQ = AC/AB ⇒ 4 AP Q ∼ 4 ACB , ◦ 6 6 giving AP Q = ACB = 70 . A P Q E F B D C 6 6