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HMMT 十一月 2013 · 团队赛 · 第 4 题

HMMT November 2013 — Team Round — Problem 4

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 4 ] Consider triangle ABC with side lengths AB = 4, BC = 7, and AC = 8. Let M be the midpointof segment AB , and let N be the point on the interior of segment AC that also lies on the circumcircleof triangle M BC . Compute BN .
    °
解析

英文解析

  1. [ 4 ] Consider triangle ABC with side lengths AB = 4, BC = 7, and AC = 8. Let M be the midpointof segment AB , and let N be the point on the interior of segment AC that also lies on the circumcircleof triangle M BC . Compute BN .
    √ √
    2 2 2
    210 105 4 +8 − 7 4
    √
    Answer: OR Let ∠ BAC = θ . Then, cos θ = . Since AM = = 2, and power
    4 2 · 4 · 8 2
    2 2
    2 · 4
    of a point gives AM · AB = AN · AC , we have AN = = 1, so N C = 8 − 1 = 7. Law of cosines ontriangle BAN gives 8
    2 2 2
    4 + 8 − 7 16 + 15 15 105
    2 2 2
    BN = 4 + 1 − 2 · 4 · 1 · = 17 − = 15 − = ,
    2 · 4 · 8 8 8 8
    √
    so BN = .210
    °4