[tex]\it \widehat{AOC} =\dfrac{\ 80^{(20}}{100}\cdot \widehat{BOC}=\dfrac{4}{5}\cdot \widehat{BOC}\ \ \ \ \ \ (1)\\ \\ \\ \widehat{AOB}=\widehat{AOC}+\widehat{BOC}=45^o \stackrel{(1)}{\Longrightarrow}\ \dfrac{4}{5}\cdot\widehat{BOC}+\widehat{BOC}=45^o|_{:5} \Rightarrow\\ \\ \\ \Rightarrow 4\cdo\widehat{BOC}+5\cdot \widehat{BOC}=45\cdot5 \Rightarrow 9\cdo\widehat{BOC}=45\cdot5|_{:9} \Rightarrow \widehat{BOC} =25^o\ \ \ \ \ (2)[/tex]
[tex]\it (1),\ (2) \Rightarrow \widehat{AOC}=\dfrac{4}{5}\cdot 25^o=20^o[/tex]
[tex]\it\ \ [OB -\ \ bisectoare\ pentru\ \widehat{COP} \Rightarrow \widehat{BOP}=\widehat{BOC} = 25^o[/tex]
[tex][OA -\ bisectoare\ pentru\ \widehat{COM} \Rightarrow \widehat{AOM} = \widehat{AOC} = 20^o[/tex]
[tex]\it \widehat{MOP} = 20^o + 20^o + 25^o + 25^o = 90^o \Rightarrow OM\ \perp\ OP[/tex]
