HMMT 二月 2020 · 冲刺赛 · 第 13 题

HMMT February 2020 — Guts Round — Problem 13

[8] Let 4 ABC be a triangle with AB = 7, BC = 1, and CA = 4 3. The angle trisectors of C intersect AB at D and E , and lines AC and BC intersect the circumcircle of 4 CDE again at X and Y , respectively. Find the length of XY .

解析

[8] Let 4 ABC be a triangle with AB = 7, BC = 1, and CA = 4 3. The angle trisectors of C intersect AB at D and E , and lines AC and BC intersect the circumcircle of 4 CDE again at X and Y , respectively. Find the length of XY . Proposed by: Hahn Lheem 112 Answer: 65 Solution: Let O be the cirumcenter of 4 CDE . Observe that 4 ABC ∼ 4 XY C . Moreover, 4 ABC √ 2 2 2 is a right triangle because 1 + (4 3) = 7 , so the length XY is just equal to 2 r , where r is the radius of the circumcircle of 4 CDE . Since D and E are on the angle trisectors of angle C , we see √ 4 3 that 4 ODE , 4 XDO , and 4 Y EO are equilateral. The length of the altitude from C to AB is . 7 √ √ XY 4 3 2 r 4 3 The distance from C to XY is · = · , while the distance between lines XY and AB is AB 7 7 7 √ r 3 . Hence we have 2 √ √ √ 4 3 2 r 4 3 r 3 = · + . 7 7 7 2 56 112 Solving for r gives that r = , so XY = . 65 65 C X Y O A D E B