1989 APMO 第 3 题

Let $A_{1}, A_{2}, A_{3}$ be three points in the plane, and for convenience,let $A_{4}=A_{1}, A_{5}=A_{2}$. For $n=1,2$, and 3 , suppose that $B_{n}$ is the midpoint of $A_{n} A_{n+1}$, and suppose that $C_{n}$ is the midpoint of $A_{n} B_{n}$. Suppose that $A_{n} C_{n+1}$ and $B_{n} A_{n+2}$ meet at $D_{n}$, and that $A_{n} B_{n+1}$ and $C_{n} A_{n+2}$ meet at $E_{n}$. Calculate the ratio of the area of triangle $D_{1} D_{2} D_{3}$ to the area of triangle $E_{1} E_{2} E_{3}$.

Answer: $\frac{25}{49}$.

设$A_{1}、A_{2}、A_{3}$为平面上的三个点，为方便起见，设$A_{4}=A_{1}、A_{5}=A_{2}$。对于 $n=1,2$ 和 3 ，假设 $B_{n}$ 是 $A_{n} A_{n+1}$ 的中点，并假设 $C_{n}$ 是 $A_{n} B_{n}$ 的中点。假设 $A_{n} C_{n+1}$ 和 $B_{n} A_{n+2}$ 在 $D_{n}$ 相交，并且 $A_{n} B_{n+1}$ 和 $C_{n} A_{n+2}$ 在 $E_{n}$ 相交。计算三角形 $D_{1} D_{2} D_{3}$ 的面积与三角形 $E_{1} E_{2} E_{3}$ 的面积之比。

答案：$\frac{25}{49}$。