HMMT 二月 2012 · 冲刺赛 · 第 14 题
HMMT February 2012 — Guts Round — Problem 14
题目详情
英文原题
[7] Let triangle ABC have AB= 5,BC= 6, and AC= 7, with circumcenter O. Extend ray ABto point Dsuch that BD= 5, and extend ray BCto point Esuch that OD=OE. Find CE.
解析
英文解析
- [ 7 ] Let triangle ABC have AB = 5, BC = 6, and AC = 7, with circumcenter O . Extend ray AB topoint 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.
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