HMMT 二月 2008 · TEAM1 赛 · 第 13 题

英文解析

13. [ 40 ] Let M be the midpoint of BC , and T diametrically opposite to D on the incircle of
    ABC . Show that DT, AM, EF are concurrent.
    Solution: If AB = AC , then the result is clear as AM and DT coincide. So, assume that
    AB 6 = AC .
    X Z YFTA
    B CIE
    M D
    Let lines DT and EF meet at Z . Construct a line through Z parallel to BC , and let it
    ° °
    meet AB and AC at X and Y , respectively. We have ∠ XZI = 90 , and ∠ XF I = 90 .
    Therefore, F, Z, I, X are concyclic, and thus ∠ IXZ = ∠ IF Z . By similar arguments, we alsohave ∠ IY Z = ∠ IEZ . Thus, triangles IF E and IXY are similar. Since IE = IF , we mustalso have IX = IY . Since IZ is an altitude of the isosceles triangle IXY , Z is the midpointof XY .
    Since XY and BC are parallel, there is a dilation centered at A that sends XY to BC . So 5
    it must send the midpoint Z to the midpoint M . Therefore, A, Z, M are collinear. It followsthat DT, AM, EF are concurrent.