HMMT 二月 2006 · 几何 · 第 10 题

HMMT February 2006 — Geometry — Problem 10

Triangle ABC has side lengths AB = 65, BC = 33, and AC = 56. Find the radius of the circle tangent to sides AC and BC and to the circumcircle of triangle ABC .

解析

Triangle ABC has side lengths AB = 65, BC = 33, and AC = 56. Find the radius of the 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