HMMT 二月 2003 · 几何 · 第 9 题
HMMT February 2003 — Geometry — Problem 9
题目详情
英文原题
- 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 circleinstersects AB again at P . Another circle is tangent to BC at C and is also tangentto segment AD ; this circle intersects AC again at Q . Find AP Q (in degrees).6
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解析
英文解析
- 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 circleinstersects AB again at P . Another circle is tangent to BC at C and is also tangentto segment AD ; this circle intersects AC again at Q . Find AP Q (in degrees).6
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,
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AP · AB = AE = AF = AQ · AC ⇒ AP/AQ = AC/AB ⇒ 4 AP Q ∼ 4 ACB ,
°
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giving AP Q = ACB = 70 .
EQPA
B D CF
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