HMMT 二月 2006 · 几何 · 第 10 题

英文解析

10. Triangle ABC has side lengths AB = 65, BC = 33, and AC = 56. Find the radius ofthe circle tangent to sides AC and BC and to the circumcircle of triangle ABC .
    Answer: 24
    Solution: Let Γ be the circumcircle of triangle ABC , and let E be the center of
    °
    the circle tangent to Γ and the sides AC and BC . Notice that ∠ C = 90 because
    2 2 2
    33 + 56 = 65 . Let D be the second intersection of line CE with Γ, so that D is
    °
    the midpoint of the arc AB away from C . Because ∠ BCD = 45 , one can easily
    √
    calculate CD = 89 2 / 2. The power of E with respect to Γ is both r (65 − r ) and
    √ √ √
    r 2 · (89 2 / 2 − r 2) = r (89 − 2 r ) , so r = 89 − 65 = 24. 3