If ODEI is a square, where O and I stand for circumcentre and incentre respectively and D and E are the points of perpendicular from O and I on the base BC, then

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ODEI is a square, so O D = O I . O I = R 2 − 2 R r ​ and O D = R cos A , so R 2 − 2 R r = R 2 cos 2 A , i.e. 1 − cos 2 A = R 2 r ​ . With cos A = R r ​ : 1 − ( R r ​ ) 2 = R 2 r ​ , or ( R r ​ ) 2 + R 2 r ​ − 1 = 0 , so R r ​ = 2 ​ − 1

Mathematics - Properties of Triangle Question with Solution | TestHub

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