HMMT 二月 2012 · 冲刺赛 · 第 14 题
HMMT February 2012 — Guts Round — Problem 14
题目详情
- [ 7 ] Let triangle ABC have AB = 5, BC = 6, and AC = 7, with circumcenter O . Extend ray AB to point D such that BD = 5, and extend ray BC to point E such that OD = OE . Find CE . 2 2
解析
- [ 7 ] Let triangle ABC have AB = 5, BC = 6, and AC = 7, with circumcenter O . Extend ray AB to point D such that BD = 5, and extend ray BC to point E such that OD = OE . Find CE . √ Answer: 59 − 3 Because OD = OE , D and E have equal power with respect to the circle, so ( EC )( EB ) = ( DB )( DA ) = 50. Letting EC = x , we have x ( x + 6) = 50, and taking the positive root √ gives x = 59 − 3. 2 2