1982 USAMO 第 3 题

If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that

$I.Q. (A_1BC) > I.Q.(A_2BC)$,

where the isoperimetric quotient of a figure $F$ is defined by

$$

I.Q.(F) = \frac{\text{Area (F)}}{\text{[Perimeter (F)]}^2}

$$

若点$A_1$在等边三角形$ABC$的内部，点$A_2$在$\triangle{A_1BC}$的内部，证明

$智商。 (A_1BC) > 智商(A_2BC)$,

其中图形 $F$ 的等周商定义为

$$

I.Q.(F) = \frac{\text{面积 (F)}}{\text{[周长 (F)]}^2}

$$