\[\boxed{\mathbf{835.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[Рисунок\ по\ условию\ задачи:\]
\[\mathbf{Дано:}\]
\[ABCD - трапеция;\]
\[\text{AD} \parallel BC;\ \ AD > BC;\]
\[E;F \in AD;\]
\[\text{BE} \parallel CF;\ \ \]
\[O = AC \cap BD;\]
\[G = AC \cap BE;\ \]
\[H = BD \cap CF.\]
\[\mathbf{Доказать:}\]
\[S_{\text{EGOHF}} = S_{\text{ABG}} + S_{\text{BCO}} + S_{\text{DCH}}.\]
\[\mathbf{Доказательство.}\]
\[1)\ S_{\text{ABC}} = S_{\text{BCD}} = \frac{1}{2}BC \bullet h.\]
\[2)\ S_{\text{ABG}} + S_{\text{BGO}} + S_{\text{BOC}} =\]
\[= S_{\text{DCH}} + S_{\text{COH}} + S_{\text{BOC}}.\]
\[3)\ S_{\text{EBCF}} = BC \bullet h = 2S_{\text{ABC}} =\]
\[= S_{\text{ABC}} + S_{\text{BCD}}.\]
\[4)\ S_{\text{EGOHF}} + S_{\text{BGO}} + S_{\text{BOC}} + S_{\text{COH}} =\]
\[5)\ S_{\text{EGOHF}} = S_{\text{ABG}} + S_{\text{BCO}} + S_{\text{DCH}}.\]
\[\mathbf{Что\ и\ требовалось\ доказать.}\]