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HMMT 二月 2007 · 几何 · 第 8 题

HMMT February 2007 — Geometry — Problem 8

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 6 ] ABCD is a convex quadrilateral such that AB < AD . The diagonal AC bisects ∠ BAD , and
    ° °
    m ∠ ABD = 130 . Let E be a point on the interior of AD , and m ∠ BAD = 40 . Given that BC =
    CD = DE , determine m ∠ ACE in degrees.
解析

英文解析

  1. [ 6 ] ABCD is a convex quadrilateral such that AB < AD . The diagonal AC bisects ∠ BAD , and
    ° °
    m ∠ ABD = 130 . Let E be a point on the interior of AD , and m ∠ BAD = 40 . Given that BC =
    CD = DE , determine m ∠ ACE in degrees.
    ° ′2
    Answer: 55 . First, we check that ABCD is cyclic. Reflect B over AC to B on AD , and note that
    ′ ′ ′ ° ′ °
    B C = CD . Therefore, m ∠ ADC = m ∠ B DC = m ∠ CB D = 180 − m ∠ AB C = 180 − m ∠ CBA .
    ° ° °
    Now m ∠ CBD = m ∠ CAD = 20 and m ∠ ADC = 180 − m ∠ CBA = 30 . Triangle CDE is isosceles,
    ° ° ° °
    so m ∠ CED = 75 and m ∠ AEC = 105 . It follows that m ∠ ECA = 180 − m ∠ AEC − m ∠ CAE = 55 .