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HMMT 二月 2005 · 冲刺赛 · 第 21 题

HMMT February 2005 — Guts Round — Problem 21

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [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

英文原题

[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

英文解析

  1. 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 , andextend AD to intersect ω again at E . Note that A subtends a quarter of the circle, soin 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.
    B7
    CDAE