HMMT 二月 2008 · 几何 · 第 4 题

HMMT February 2008 — Geometry — Problem 4

[ 4 ] In a triangle ABC , take point D on BC such that DB = 14 , DA = 13 , DC = 4, and the circumcircle of ADB is congruent to the circumcircle of ADC . What is the area of triangle ABC ? ◦ ◦

解析

[ 4 ] In a triangle ABC , take point D on BC such that DB = 14 , DA = 13 , DC = 4, and the circumcircle of ADB is congruent to the circumcircle of ADC . What is the area of triangle ABC ? Answer: 108 A B C M D The fact that the two circumcircles are congruent means that the chord AD must subtend the same angle in both circles. That is, ∠ ABC = ∠ ACB , so ABC is isosceles. Drop the perpendicular M from A to BC ; we know M C = 9 and so M D = 5 and by Pythagoras on AM D , AM = 12. Therefore, the 1 1 area of ABC is ( AM )( BC ) = (12)(18) = 108. 2 2