HMMT 二月 2005 · 冲刺赛 · 第 21 题

HMMT February 2005 — Guts Round — Problem 21

[8] In triangle ABC with altitude AD , ∠ BAC = 45 , DB = 3, and CD = 2. Find the area of triangle ABC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, FEBRUARY 19, 2005 — GUTS ROUND

解析

In triangle ABC with altitude AD , ∠ BAC = 45 , DB = 3, and CD = 2. Find the area of triangle ABC . Solution: 15 Suppose first that D lies between B and C . Let ABC be inscribed in circle ω , and extend AD to intersect ω again at E . Note that A subtends a quarter of the circle, so in particular, the chord through C perpendicular to BC and parallel to AD has length BC = 5. Therefore, AD = 5 + DE . By power of a point, 6 = BD · DC = AD · DE = 2 1 AD − 5 AD , implying AD = 6, so the area of ABC is BC · AD = 15. 2 √ If D does not lie between B and C , then BC = 1, so A lies on a circle of radius 2 / 2 through B and C . But then it is easy to check that the perpendicular to BC through D cannot intersect the circle, a contradiction. 7 B E A D C