I4.1 In Figure $1, A B C D$ is a square, $E$ is a point and $\angle E A B=30^{\circ}$. If the area of $A B C D$ is six times that of $\triangle A B E$, then the ratio of $A E: A B=$ $a: 1$. Find the value of $a$.

Remark: The original questions is
在圖一中, $A B C D$ 為一正方形, $E$ 為此正方形以外的一點及 $\angle E A B=30^{\circ}$ 。若 $A B C D$ 的面積是 $\triangle A B E$ 的面積的六倍, 則 $A E: A B=a: 1$ 。求 $a$ 的值。
In Figure $1, A B C D$ is a square, $E$ is a point outside the square and $\angle E A B=30^{\circ}$. If the area of $A B C D$ is six times that of $\triangle A B E$, then the ratio of $A E: A B=a: 1$. Find the value of $a$.
In fact, $E$ must lie inside the square.
Let $A B=A D=1, A E=a$, let the altitude of $\triangle A B E$ from $E$ to $A B$ be $h$. area of $A B C D$ is six times that of $\triangle A B E \Leftrightarrow 1^{2}=6 \times \frac{1}{2} \cdot 1 \cdot h$
$$
h=\frac{1}{3}, a=\frac{h}{\sin 30^{\circ}}=2 h=\frac{2}{3}
$$