Монгол Бодлогын Сан

Заавар: $$S_{\triangle FED}=S_{\triangle ABC}-S_{\triangle AFD}-S_{\triangle BFE}-S_{\triangle CED}$$ болохыг ашигла.

Бодолт: $AB=7x$, $AC=7y$, $BC=7z$ гэвэл $$\dfrac{AF}{FB}=\dfrac{BE}{EC}=\dfrac{CD}{AD}=\dfrac{2}{5}$$ тул $AF=2x$, $FB=5x$; $DC=2y$, $AD=5y$; $BE=2z$, $EC=5z$ байна. Нөгөө талаас $$S=\dfrac12\cdot 49xy\sin\alpha=\dfrac12\cdot 49xz\sin\beta=\dfrac12\cdot 49yz\sin\gamma$$ тул $$\begin{align*} S_{\triangle AFD}&=\dfrac12\cdot 2x\cdot 5y\sin\alpha=5xy\cdot\dfrac{2S}{49xy}=\dfrac{10}{49}S\\ S_{\triangle BFE}&=\dfrac12\cdot 5x\cdot 2z\sin\beta=5xz\cdot\dfrac{2S}{49xz}=\dfrac{10}{49}S\\ S_{\triangle CED}&=\dfrac12\cdot 5z\cdot 2y\sin\gamma=5yz\cdot\dfrac{2S}{49yz}=\dfrac{10}{49}S \end{align*}$$ Иймд $$S_{\triangle FED}=S-3\cdot\dfrac{10}{49}S=\dfrac{19}{49}S\Rightarrow\dfrac{S_{\triangle FED}}{S}=\dfrac{19}{49}$$ байна.