HMMT 二月 2007 · 冲刺赛 · 第 23 题

HMMT February 2007 — Guts Round — Problem 23

[ 12 ] In triangle ABC , ∠ ABC is obtuse. Point D lies on side AC such that ∠ ABD is right, and point E lies on side AC between A and D such that BD bisects ∠ EBC . Find CE , given that AC = 35 , BC = 7, and BE = 5.

解析

[ 12 ] In triangle ABC , ∠ ABC is obtuse. Point D lies on side AC such that ∠ ABD is right, and point E lies on side AC between A and D such that BD bisects ∠ EBC . Find CE , given that AC = 35 , BC = 7, and BE = 5. ′ ′ Answer: 10 . Reflect A and E over BD to A and E respectively. Note that the angle conditions ′ ′ ′ ′ show that A and E lie on AB and BC respectively. B is the midpoint of segment AA and CE = ′ BC − BE = 2. Menelaus’ theorem now gives ′ ′ CD AA BE · · = 1 , ′ ′ DA A B E C from which DA = 5 CD or CD = AC/ 6. By the angle bisector theorem, DE = 5 CD/ 7, so that CE = 12 CD/ 7 = 10.