HMMT 二月 2006 · 几何 · 第 3 题

HMMT February 2006 — Geometry — Problem 3

Let A , B , C , and D be points on a circle such that AB = 11 and CD = 19 . Point P is on segment AB with AP = 6, and Q is on segment CD with CQ = 7. The line through P and Q intersects the circle at X and Y . If P Q = 27, find XY .

解析

Let A , B , C , and D be points on a circle such that AB = 11 and CD = 19 . Point P is on segment AB with AP = 6, and Q is on segment CD with CQ = 7. The line through P and Q intersects the circle at X and Y . If P Q = 27, find XY . Answer: 31 Solution: Suppose X, P, Q, Y lie in that order. Let P X = x and QY = y . By power of a point from P , x · (27 + y ) = 30, and by power of a point from Q , y · (27 + x ) =
Subtracting the first from the second, 27 · ( y − x ) = 54, so y = x + 2. Now, x · (29 + x ) = 30, and we find x = 1 , − 30. Since − 30 makes no sense, we take x = 1 and obtain XY = 1 + 27 + 3 = 31 .