HMMT 二月 2005 · 几何 · 第 10 题

英文解析

10. Let AB be the diameter of a semicircle Γ. Two circles, ω and ω , externally tangent
    1 2
    to each other and internally tangent to Γ, are tangent to the line AB at P and Q ,
    respectively, and to semicircular arc AB at C and D , respectively, with AP < AQ .
    °
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    Suppose F lies on Γ such that F QB = CQA and that ABF = 80 . Find P DQin degrees.
    Solution: 35
    Extend the semicircle centered at O to an entire circle ω , and let the reflection of F
    ′ ′
    over AB be F . Then CQF is a straight line. Also, the homothety centered at Ctaking ω into ω takes P to a point X on ω and AB to the parallel line tangent to ωat X . Therefore, X is the midpoint of semicircle AXB , and C , P , and X lie on a line.1
    Similarly, D , Q , and X lie on a line. So,
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    45 = XCB = P CB = P CQ + QCB = P CQ + 10 ,
    ′ ′ ° ° °
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    since QCB = F CB = F AB = F AB = 90 − ABF = 10 . Thus P CQ = 35 .
    °
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    We will show that P CQ = P DQ to get that P DQ = 35 .
    Note that XP Q subtends the sum of arcs AC and BX , which is equal to arc XC .6
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    Therefore XP Q = CDX , so CDQP is cyclic and P CQ = P DQ . The conclusionfollows.
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