PUMaC 2015 · 几何（B 组） · 第 7 题

PUMaC 2015 — Geometry (Division B) — Problem 7

[ 7 ] Let P, A, B, C be points on circle O such that C does not lie on arc BAP and P A = ◦ 21 , P B = 56 , P C = 35 and m ∠ BP C = 60 . Now choose point D on the circle such that C does É not lie on arc BDP and BD = 39. What is length AD ? Due to spacing, last problem is on the next page. 1

解析

[ 7 ] Let P, A, B, C be points on circle O such that C does not lie on arc BAP and P A = ◦ 21 , P B = 56 , P C = 35 and m ∠ BP C = 60 . Now choose point D on the circle such that C does É not lie on arc BDP and BD = 39. What is length AD ? 3 Solution: 16 ◦ First, using the Law of Cosines on ∆ BP C , we get that BC = 49. Now m ∠ BOC = 2 ⋅ 60 = 120 ′ ′ and there is a point A on the circle such that ∆ A BC is an equilateral triangle. Then since ′ Ç P B = 56 > 49, we know that P is on minor arc A C . If we apply Ptolemy’s theorem on the ′ cyclic quadrilateral P A BC , we get that: ′ ′ ′ P A ⋅ BC + P C ⋅ A B = P B ⋅ A C ′ ′ É Plugging in numbers gives P A = 21 and since A lies on the arc BAP , we know that A and ′ A are the same point and hence △ ABC is equilateral. 2 2 2 ◦ Now using Law of Cosines again, we get that AD + BD − 2 AD ⋅ BD cos 120 = AB and so AD = 16.