HMMT 十一月 2011 · 冲刺赛 · 第 22 题

英文解析

22. [ 12 ] Let ABC be a triangle with AB = 23, BC = 24, and CA = 27. Let D be the point on segment
    AC such that the incircles of triangles BAD and BCD are tangent. Determine the ratio CD/DA .
    Answer: Let X, Z, E be the points of tangency of the incircle of ABD to AB, BD, DA respec-14
    tively. Let Y, Z, F be the points of tangency of the incircle of CBD to CB, BD, DC respectively. We 13
    note that
    CB + BD + DC = CY + Y B + BZ + ZD + DF + F C = 2( CY ) + 2( BY ) + 2( DF )2(24) + 2( DF )
    by equal tangents, and that similarly
    AB + BD + DA = 2(23) + 2( DE ) .
    Since DE = DZ = DF by equal tangents, we can subtract the equations above to get that
    CB + CD − AB − AD = 2(24) − 24(23) ⇒ CD − DA = 1 .
    Since we know that CD + DA = 27, we get that CD = 14, DA = 13, so the desired ratio is .14
    Guts Round 13
    ZXYB
    A E D F C