HMMT 二月 2007 · 几何 · 第 4 题

HMMT February 2007 — Geometry — Problem 4

[ 4 ] Circle ω has radius 5 and is centered at O . Point A lies outside ω such that OA = 13. The two tangents to ω passing through A are drawn, and points B and C are chosen on them (one on each tangent), such that line BC is tangent to ω and ω lies outside triangle ABC . Compute AB + AC given that BC = 7.

解析

[ 4 ] Circle ω has radius 5 and is centered at O . Point A lies outside ω such that OA = 13. The two tangents to ω passing through A are drawn, and points B and C are chosen on them (one on each tangent), such that line BC is tangent to ω and ω lies outside triangle ABC . Compute AB + AC given that BC = 7. 1 Answer: 17 . Let T , T , and T denote the points of tangency of AB, AC, and BC with ω , respec- 1 2 3 √ 2 2 tively. Then 7 = BC = BT + T C = BT + CT . By Pythagoras, AT = AT = 13 − 5 = 12. 3 3 1 2 1 2 Now note that 24 = AT + AT = AB + BT + AC + CT = AB + AC + 7. 1 2 1 2