HMMT 二月 2022 · 冲刺赛 · 第 23 题

HMMT February 2022 — Guts Round — Problem 23

[12] Let ABCD be an isosceles trapezoid such that AB = 17, BC = DA = 25, and CD = 31. Points P and Q are selected on sides AD and BC , respectively, such that AP = CQ and P Q = 25. Suppose that the circle with diameter P Q intersects the sides AB and CD at four points which are vertices of a convex quadrilateral. Compute the area of this quadrilateral.

解析

[12] Let ABCD be an isosceles trapezoid such that AB = 17, BC = DA = 25, and CD = 31. Points P and Q are selected on sides AD and BC , respectively, such that AP = CQ and P Q = 25. Suppose that the circle with diameter P Q intersects the sides AB and CD at four points which are vertices of a convex quadrilateral. Compute the area of this quadrilateral. Proposed by: Fedir Yudin Answer: 168 Solution: ′ ′ Let the midpoint of P Q be M ; note that M lies on the midline of ABCD . Let B C be a translate of ′ ′ ′ ′ BC (parallel to AB and CD ) so that M is the midpoint of B and C . Since M B = M C = 25 / 2 = ′ ′ M P = M Q , B and C are one of the four intersections of the circle with diameter P Q and the sides ′ ′ AB and CD . We may also define A and D similarly and get that they are also among the four points. ′ ′ ′ ′ It follows that the desired quadrilateral is B D C A , which is a rectangle with height equal to the 1 height of ABCD (which is 24), and width equal to (31 − 17) = 7. Thus the area is 24 · 7 = 168. 2