HMMT 二月 2007 · 几何 · 第 4 题
HMMT February 2007 — Geometry — Problem 4
题目详情
英文原题
- [ 4 ] Circle ω has radius 5 and is centered at O . Point A lies outside ω such that OA = 13. The twotangents to ω passing through A are drawn, and points B and C are chosen on them (one on eachtangent), such that line BC is tangent to ω and ω lies outside triangle ABC . Compute AB + AC giventhat BC = 7.
解析
英文解析
- [ 4 ] Circle ω has radius 5 and is centered at O . Point A lies outside ω such that OA = 13. The twotangents to ω passing through A are drawn, and points B and C are chosen on them (one on eachtangent), such that line BC is tangent to ω and ω lies outside triangle ABC . Compute AB + AC giventhat BC = 7.
Answer: 17 . Let T , T , and T denote the points of tangency of AB, AC, and BC with ω , respec-1
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tively. Then 7 = BC = BT + T C = BT + CT . By Pythagoras, AT = AT = 13 − 5 = 12.
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Now note that 24 = AT + AT = AB + BT + AC + CT = AB + AC + 7.
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