HMMT 十一月 2014 · 冲刺赛 · 第 24 题
HMMT November 2014 — Guts Round — Problem 24
题目详情
英文原题
- [ 12 ] Let ABC be a triangle with AB = 13, BC = 14, and CA = 15. We construct isosceles right
°
triangle ACD with ∠ ADC = 90 , where D, B are on the same side of line AC , and let lines AD and
°
CB meet at F . Similarly, we construct isosceles right triangle BCE with ∠ BEC = 90 , where E, Aare on the same side of line BC , and let lines BE and CA meet at G . Find cos ∠ AGF .
HMMT NOVEMBER 2014, 15 NOVEMBER 2014 — GUTS ROUND
Organization Team Team ID#
解析
英文解析
- [ 12 ] Let ABC be a triangle with AB = 13, BC = 14, and CA = 15. We construct isosceles right
°
triangle ACD with ∠ ADC = 90 , where D, B are on the same side of line AC , and let lines AD and
°
CB meet at F . Similarly, we construct isosceles right triangle BCE with ∠ BEC = 90 , where E, Aare on the same side of line BC , and let lines BE and CA meet at G . Find cos ∠ AGF .
°5
Answer: − We see that ∠ GAF = ∠ GBF = 45 , hence quadrilateral GF BA is cyclic.
°13
Consequently ∠ AGF + ∠ F BA = 180 . So cos ∠ AGF = − cos ∠ F BA . One can check directly thatcos ∠ CBA = (say, by the Law of Cosines).5
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