Question

In a triangle $$ABC$$, the line joining the circumcentre to the incentre is parallel to $$BC$$, then evaluate $$\cos { B } +\cos { C }$$

Solution

In $$\triangle OBL, \displaystyle \cos { A } =\frac { OL }{ OB } =\frac { r }{ R }$$$$\Rightarrow R\cos { A } =r$$$$\displaystyle \Rightarrow R\cos { A } =4R\sin { \frac { A }{ 2 } } \sin { \frac { B }{ 2 } } \sin { \frac { C }{ 2 } }$$$$\displaystyle \Rightarrow \cos { A } =4\sin { \frac { A }{ 2 } } \sin { \frac { B }{ 2 } } \sin { \frac { C }{ 2 } }$$$$\Rightarrow \cos { A } =\cos { A } +\cos { B } +\cos { C } -1$$$$\Rightarrow 0=\cos { B } +\cos { C } -1\Rightarrow \cos { B } +\cos { C } =1$$Mathematics

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