HMMT 二月 2006 · 几何 · 第 6 题

英文解析

6. A circle of radius t is tangent to the hypotenuse, the incircle, and one leg of an isoscelesπright triangle with inradius r = 1 + sin . Find rt .
   √8
   2 + 2
   Answer:
   Solution: The distance between the point of tangency of the two circles and the 4
   π πnearest vertex of the triangle is seen to be both r (csc − 1) and t (csc + 1), so
   8 8
   π π π
   2 2
   r (csc − 1) (1 + sin ) (1 − sin )
   2π
   8 8 8
   rt = = = 1 − sinπ πcsc + 1 1 + sin 8
   8 8
   √ √
   2 π π
   1 − 2 sin cos
   1 1 1 2 2 + 2
   8 4 = + = + = + = .
   2 2 2 2 2 4 4