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Point on hypotenuse making the two inradii equal

Source: INMO 2016 Problem 5

January 17, 2016
geometryinradiustrigonometryequation

Problem Statement

Let ABCABC be a right-angle triangle with ∠B=90∘\angle B=90^{\circ}. Let DD be a point on ACAC such that the inradii of the triangles ABDABD and CBDCBD are equal. If this common value is r′r^{\prime} and if rr is the inradius of triangle ABCABC, prove that 1r′=1r+1BD. \cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}.
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