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HMMT 二月 2012 · 几何 · 第 3 题

HMMT February 2012 — Geometry — Problem 3

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

Let ABC be a triangle with incenter I. Let the circle centered at Band passing through Iintersect sideABatDand let the circle centered at Cpassing through Iintersect side ACatE. Suppose DE is the perpendicular bisector of AI. What are all possible measures of angle BAC in degrees?

解析

英文解析

  1. Let ABC be a triangle with incenter I . Let the circle centered at B and passing through I intersectside AB at D and let the circle centered at C passing through I intersect side AC at E . Suppose DEis the perpendicular bisector of AI . What are all possible measures of angle BAC in degrees?
    Answer: Let α = ∡ BAC . DE is the perpendicular bisector of AI , so DA = DI , and 540
    °7
    ∠ DIA = ∠ DAI = α/ 2. Thus, ∠ IDB = ∠ DIB = α , since BD = BI . This gives ∠ DBI = 180 − 2 α ,
    ° °
    so that ∠ ABC = 360 − 4 α . Similarly, ∠ ACB = 360 − 4 α . Now, summing the angles in ABC , we
    °
    ° °540
    find 720 − 7 α = 180 , so that α = . 7