Question 121
Let $\triangle ABC$ be a triangle. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, and point $F$ is on line segment $DE$. Let $\frac{AD}{AB} = x$, $\frac{AE}{AC} = y$, $\frac{DF}{DE} = z$. Prove that
(1) $S_{\triangle BDF} = (1-x)y zS_{\triangle ABC}$ and $S_{\triangle CEF} = x(1-y)(1-z)S_{\triangle ABC}$;
(2) $\sqrt[3]{S_{\triangle BDF}} + \sqrt[3]{S_{\triangle CEF}} \le \sqrt[3]{S_{\triangle ABC}}$.
(posed by Li Weigu)
(1) $S_{\triangle BDF} = (1-x)y zS_{\triangle ABC}$ and $S_{\triangle CEF} = x(1-y)(1-z)S_{\triangle ABC}$;
(2) $\sqrt[3]{S_{\triangle BDF}} + \sqrt[3]{S_{\triangle CEF}} \le \sqrt[3]{S_{\triangle ABC}}$.
(posed by Li Weigu)