HMMT 二月 2013 · 几何 · 第 10 题

HMMT February 2013 — Geometry — Problem 10

Triangle ABC is inscribed in a circle ω . Let the bisector of angle A meet ω at D and BC at E . Let ′ ′ ◦ the reflections of A across D and C be D and C , respectively. Suppose that ∠ A = 60 , AB = 3, and ′ AE = 4. If the tangent to ω at A meets line BC at P , and the circumcircle of AP D meets line BC ′ at F (other than P ), compute F C . HMMT 2013 Saturday 16 February 2013 Geometry Test PUT LABEL HERE Name Team ID# Organization Team

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解析

Triangle ABC is inscribed in a circle ω . Let the bisector of angle A meet ω at D and BC at E . Let ′ ′ ◦ the reflections of A across D and C be D and C , respectively. Suppose that ∠ A = 60 , AB = 3, and ′ AE = 4. If the tangent to ω at A meets line BC at P , and the circumcircle of AP D meets line BC ′ at F (other than P ), compute F C . √ √ 1 Answer: 2 13 − 6 3 First observe that by angle chasing, ∠ P AE = 180 − ∠ BAC − ∠ ABC = 2 ′ ′ ′ ′ ∠ AEP , so by the cyclic quadrilateral AP D F , ∠ EF D = ∠ P AE = ∠ P EA = ∠ D EF . Thus, ED F is isosceles. ′ ′ ′ ′ ′ ′ Define B to be the reflection of A about B , and observe that B C || EF and B D C is isosceles. It ′ ′ ′ ′ follows that B EF C is an isosceles trapezoid, so F C = B E , which by the law of cosines, is equal to √ √ √ ′ 2 2 AB + AE − 2 AB · AE cos 30 = 2 13 − 6 3. Geometry Test