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第31届美国大学生数学竞赛B卷第6题是一道很有趣的题目,本文用命题的形式表述如下:命题1边长依次为a,b,c,d的圆外切四边形的面积S=abcd~(1/2),则它可内接于某圆.此命题构思精妙,但其证明不好入手,经笔者探究后发现,此命题表述为充分必要条件时仍然成立,而且通过充分性的证明,可以很自然的找到必要性(命题1)的证明思路.现把命题1的结论改为充分必要条件后得到如下命题:
The 31st American College Students Mathematical Contest Volume B, Volume 6 is an interesting topic. The paper uses the form of propositions to describe the following: Proposition 1 Area length of circumscribed quadrilaterals of a, b, c, d in succession S = abcd ~ (1/2), then it can be inscribed in a circle.This proposition is subtle, but its proof is not good enough. After the author has probed it, it is found that this proposition still holds when it is necessary and sufficient, Prove that it is natural to find the proof of necessity (Proposition 1). Now the proposition of Proposition 1 is replaced by the necessary and sufficient conditions to get the following proposition: