HMMT 二月 2013 · 几何 · 第 7 题
HMMT February 2013 — Geometry — Problem 7
题目详情
英文原题
Let ABC be an obtuse triangle with circumcenter Osuch that ∠ABC = 15◦and∠BAC > 90◦. Suppose that AOmeets BC atD, and that OD2+OC·DC =OC2. Find ∠C.
解析
英文解析
- Let ABC be an obtuse triangle with circumcenter O such that ∠ ABC = 15 and ∠ BAC > 90 .
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Suppose that AO meets BC at D , and that OD + OC · DC = OC . Find ∠ C .
Answer: 35 Let the radius of the circumcircle of △ ABC be r .
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OD + OC · CD = OC
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OC · CD = OC − OD
OC · CD = ( OC + OD )( OC − OD )
OC · CD = ( r + OD )( r − OD )
By the power of the point at D,
OC · CD = BD · DCr = BD
Then, △ OBD and △ OAB and △ AOC are isosceles triangles. Let ∠ DOB = α . ∠ BAO = 90 − . Inαα2
△ ABD , 15 + 90 − = α . This means that α = 70. Furthermore, ∠ ACB intercepts minor arc AB ,
∠ AOB 702
thus ∠ ACB = = = 35
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