1989 Putnam B5

Label the vertices of a trapezoid $T$ (quadrilateral with two parallel sides)

inscribed in the unit circle as $A,\,B,\,C,\,D$ so that $AB$ is parallel to

$CD$ and $A,\,B,\,C,\,D$ are in counterclockwise order. Let

$s_1,\,s_2$, and $d$ denote the lengths of the line segments

$AB,\, CD$, and $OE$, where E is the point of intersection of the diagonals

of $T$, and $O$ is the center of the circle. Determine the least upper bound of

$\frac{s_1-s_2}{d}$ over all such $T$ for which $d\ne 0$, and describe all

cases, if any, in which it is attained.

标记梯形 $T$(有两条平行边的四边形)的顶点

内接于单位圆 $A,\,B,\,C,\,D$ 使得 $AB$ 平行于

$CD$ 和 $A,\,B,\,C,\,D$ 按逆时针顺序排列。让

$s_1、\、s_2$ 和 $d$ 表示线段的长度

$AB、\、CD$ 和 $OE$，其中 E 是对角线的交点

$T$ 为圆心，$O$ 为圆心。确定最小上界

$\frac{s_1-s_2}{d}$ 超过所有这样的 $T$，其中 $d\ne 0$，并描述所有

达到该目标的情况(如果有)。