HMMT 十一月 2019 · 团队赛 · 第 8 题

HMMT November 2019 — Team Round — Problem 8

[60] In 4 ABC , the external angle bisector of ∠ BAC intersects line BC at D . E is a point on ray AC DB such that ∠ BDE = 2 ∠ ADB . If AB = 10, AC = 12, and CE = 33, compute . DE

解析

[60] In 4 ABC , the external angle bisector of ∠ BAC intersects line BC at D . E is a point on ray AC DB such that ∠ BDE = 2 ∠ ADB . If AB = 10, AC = 12, and CE = 33, compute . DE Proposed by: Milan Haiman 2 Answer: 3 − → Let F be a point on ray CA such that ∠ ADF = ∠ ADB . 4 ADF and 4 ADB are congruent, so AF = 10 and DF = DB . So, CF = CA + AF = 22. Since ∠ F DC = 2 ∠ ADB = ∠ EDC , by the angle bisector DF CF 22 2 theorem we compute = = = . DE CE 33 3