PUMaC 2016 · 团队赛 · 第 9 题
PUMaC 2016 — Team Round — Problem 9
题目详情
- (6) Let 4 ABC be a right triangle with AB = 4, BC = 5, and hypotenuse AC . Let I be the incenter of 4 ABC and E be the excenter of 4 ABC opposite A (the center of the circle tangent to BC and the extensions of segments AB and AC ). Suppose the circle with diameter √ IE intersects line AB beyond B at D . If BD = a − b , where a and b are positive integers. Find a + b .
解析
- Note that AC = 41. An angle chase shows that BICED is cyclic, and the reflection across √ √ AI takes D to C . Therefore AD = AC = 41, so BD = 41 − 4. Thus, the answer is 41 + 4 = 45 .