1990 AMC 12 第 28 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
28.
一个四边形的连续四条边长依次为 、、 和 。它既内接于一个圆，又有一个圆内切于它。内切圆与长度为 的边相切，切点将该边分成长度为 和 的两段。求 。
A quadrilateral that has consecutive sides of lengths and is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length divides that side into segments of lengths and Find
答案：B
小提示：
在每个顶点处设一个切线段长，使相邻的两个切线段长之和等于对应边长
Assign one tangent length to each vertex, so adjacent pairs sum to the four side lengths
大提示：
对于互补的两个对角，两个顶点处的切线段长之积相等
For supplementary opposite angles, the products of the tangent lengths at opposite vertices are equal
解答：
设从连续四个顶点引出的切线段长为 。则 因此 、、。若内切圆半径为 ，则角 所在顶点的切线段长为 。两个对角互补，所以 。因此 解得 。所以长度为 的边上的两段分别为 和 ，两者之差为 。
所以正确答案是 B。
Let the tangent lengths from the four consecutive vertices be Then Thus and If the inradius is a vertex with angle has tangent length Opposite angles are supplementary, so Therefore giving Hence the two segments of the -side are and whose difference is
Thus the correct answer is B.
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