HMMT 二月 2009 · GEN1 赛 · 第 10 题

英文解析

10. [ 6 ] A kite is a quadrilateral whose diagonals are perpendicular. Let kite ABCD be such that ∠ B =
    °
    ∠ D = 90 . Let M and N be the points of tangency of the incircle of ABCD to AB and BC respectively.
    ′ ′ ′
    Let ω be the circle centered at C and tangent to AB and AD . Construct another kite AB C D that is
    ′ ′ ′ ′
    similar to ABCD and whose incircle is ω . Let N be the point of tangency of B C to ω . If M N ‖ AC ,
    then what is the ratio of AB : BC ?
    √2
    1+ 5
    Answer:
    Solution: Let’s focus on the right triangle ABC and the semicircle inscribed in it since the situation 2
    is symmetric about AC . First we find the radius a of circle O . Let AB = x and BC = y . Drawing theradii OM and ON , we see that AM = x − a and 4 AM O ∼ 4 ABC . In other words,
    AM AB
    M O BC=
    x − a xa y=
    a = .xyx + y
    Now we notice that the situation is homothetic about A . In particular,
    ′ ′
    4 AM O ∼ 4 ON C ∼ 4 CN C .
    ′ ′ ′
    Also, CB and CN are both radii of circle C . Thus, when M N ‖ AC , we have
    ′
    AM = CN = CBx − a = ya = = x − yxyx + y
    2 2
    x − xy − y = 0
    √
    y y 2
    x = ± + y 2
    2 4
    √
    AB x 1 + 5 = = .
    BC y 2 3