5. $\triangle A B C$ has area 1. $E$ and $F$ are points on $A B$ and $A C$ respectively such that $E F / / B C$. If $\triangle A E F$ and $\triangle E B C$ have equal area, find the area of $\triangle E F C$.
$\triangle A B C$ 的面積是 $1 \circ E 、 F$ 分別是 $A B$ 和 $A C$ 上的點, 使得 $E F / / B C \circ$ 若 $\triangle A E F$ 和 $\triangle E B C$ 的面積相等, 求 $\triangle E F C$ 的面積.

The area of $\triangle A B C$ is 1. Points $E$ and $F$ are on $A B$ and $A C$ respectively such that $E F / / B C$. If the areas of $\triangle A E F$ and $\triangle E B C$ are equal, find the area of $\triangle E F C$.
$\triangle A B C$ has an area of $1 \circ E, F$ are points on $A B$ and $A C$ respectively, such that $E F / / B C \circ$ If the areas of $\triangle A E F$ and $\triangle E B C$ are equal, find the area of $\triangle E F C$.
5. $\sqrt{5}-2$
5. Using $E F$ and $B C$ as bases, let the heights of $\triangle A E F$ and $\triangle E B C$ be $k$ times and $1-k$ times that of $\triangle A B C$ respectively. Let $[X Y Z]$ denote the area of $\triangle X Y Z$. Since $\triangle A E F \sim \triangle A B C$, we have $[A E F]=k^{2}$; since $\triangle E B C$ shares the same base as $\triangle A B C$, we have $[E B C]=1-k$. Hence we get $k^{2}=1-k$, and thus $k=\frac{-1+\sqrt{5}}{2}$ as $k>0$. It follows that
$$
[E F C]=1-k^{2}-(1-k)=\sqrt{5}-2 .
$$