1972 USAMO 第 5 题

A given convex pentagon $ABCDE$ has the property that the area of each of the five triangles $ABC$, $BCD$, $CDE$, $DEA$, and $EAB$ is unity. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.

$$

[asy] size(120); defaultpen(fontsize(10)); pair A=dir(90), B=dir(90-72), C=dir(90-2*72), D=dir(90-3*72), E=dir(90-4*72); draw(A--B--C--D--E--cycle); draw(A--C--E--B--D--cycle); label("A",A,A);label("B",B,B);label("C",C,C);label("D",D,D);label("E",E,E); [/asy]

$$

给定的凸五边形 $ABCDE$ 具有以下属性：五个三角形 $ABC$、$BCD$、$CDE$、$DEA$ 和 $EAB$ 中每个三角形的面积均为 1。证明具有上述性质的所有五边形具有相同的面积，并计算该面积。此外，证明存在无限多个具有上述面积性质的非全等五边形。

$$

[asy] 大小(120);默认笔(字体大小(10))；对 A=dir(90), B=dir(90-72), C=dir(90-2*72), D=dir(90-3*72), E=dir(90-4*72);绘制(A--B--C--D--E--循环)；绘制(A--C--E--B--D--循环)；标签("A",A,A);标签("B",B,B);标签("C",C,C);标签("D",D,D);标签("E",E,E); [/asy]

$$