HMMT 二月 2013 · 几何 · 第 10 题
HMMT February 2013 — Geometry — Problem 10
题目详情
英文原题
Triangle ABC is inscribed in a circle ω. Let the bisector of angle Ameet ωatDandBC atE. Let the reflections of Aacross DandCbeD′andC′, respectively. Suppose that ∠A= 60◦,AB= 3, and AE= 4. If the tangent to ωatAmeets line BC atP, and the circumcircle of APD′meets line BC atF(other than P), compute FC′. HMMT 2013 Saturday 16 February 2013 Geometry Test PUT LABEL HERE Name Team ID# Organization Team 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Score:
解析
英文解析
- 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 .
√
√
Answer: 2 13 − 6 3 First observe that by angle chasing, ∠ P AE = 180 − ∠ BAC − ∠ ABC =1
′ ′ ′ ′2
∠ AEP , so by the cyclic quadrilateral AP D F , ∠ EF D = ∠ P AE = ∠ P EA = ∠ D EF . Thus, ED Fis 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.
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