PUMaC 2014 · 几何（A 组） · 第 4 题

PUMaC 2014 — Geometry (Division A) — Problem 4

[ 4 ] Consider the cyclic quadrilateral with sides 1, 4, 8, 7 in that order. What is its circumdi- √ ameter? Let the answer be of the form a b + c , for b square free. Find a + b + c

解析

[ 4 ] Consider the cyclic quadrilateral with sides 1, 4, 8, 7 in that order. What is its circumdi- √ ameter? Let the answer be of the form a b + c , for b square free. Find a + b + c Solution: Let the quadrilaterial be denoted by ABCD with AB = 1 , BC = 4 , CD = 8 , AD = 7. We ′ note that if we reflect point D across the perpendicular bisector of AC to D , we get the same ′ ′ 2 ′ 2 2 ′ 2 circle. But then, we have AD = 8 , CD = 7 and so AB + AD = BC + D C = 65 which √ is the circumdiameter squared. Hence the circumdiameter is 65. Thus a + b + c = 66