HMMT 二月 2015 · 冲刺赛 · 第 29 题

HMMT February 2015 — Guts Round — Problem 29

[ 20 ] Let ABC be a triangle whose incircle has center I and is tangent to BC , CA , AB , at D , E , F . ̂ Denote by X the midpoint of major arc BAC of the circumcircle of ABC . Suppose P is a point on line XI such that DP ⊥ EF . Given that AB = 14, AC = 15, and BC = 13, compute DP .

解析

[ 20 ] Let ABC be a triangle whose incircle has center I and is tangent to BC , CA , AB , at D , E , F . ̂ Denote by X the midpoint of major arc BAC of the circumcircle of ABC . Suppose P is a point on line XI such that DP ⊥ EF . Given that AB = 14, AC = 15, and BC = 13, compute DP . √ 4 5 Answer: Let H be the orthocenter of triangle DEF . We claim that P is the midpoint of DH . 5 Indeed, consider an inversion at the incicrle of ABC , denoting the inverse of a point with an asterik. ◦ ∗ ∗ ◦ It maps ABC to the nine-point circle of △ DEF . According to ∠ IAX = 90 , we have ∠ A X I = 90 . ∗ Hence line XI passes through the point diametrically opposite to A , which is the midpoint of DH , as claimed. Guts The rest is a straightforward computation. The inradius of △ ABC is r = 4. The length of EF is given AF · r 16 √ by EF = 2 = . Then, AI 5 ( ) 2 ( ) 1 1 64 16 2 2 2 2 DP = DH = 4 r − EF = 4 − = . 2 4 5 5 √ 4 5 Hence DP = . 5 Remark. This is also not too bad of a coordinate bash.