HMMT 二月 2013 · 几何 · 第 7 题

HMMT February 2013 — Geometry — Problem 7

Let ABC be an obtuse triangle with circumcenter O such that ∠ ABC = 15 and ∠ BAC > 90 . 2 2 Suppose that AO meets BC at D , and that OD + OC · DC = OC . Find ∠ C .

解析

Let ABC be an obtuse triangle with circumcenter O such that ∠ ABC = 15 and ∠ BAC > 90 . 2 2 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 . 2 2 OD + OC · CD = OC 2 2 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 · DC r = BD α Then, △ OBD and △ OAB and △ AOC are isosceles triangles. Let ∠ DOB = α . ∠ BAO = 90 − . In 2 α △ ABD , 15 + 90 − = α . This means that α = 70. Furthermore, ∠ ACB intercepts minor arc AB , 2 ∠ AOB 70 thus ∠ ACB = = = 35 2 2