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1/r_B +1/r_C= 2\cdot ( 1/r + 1/b +1/c), inradii in tirangle with <A=60

Source: Mediterranean Mathematical Olympiad 2019 P1 MMC

July 21, 2019
geometryincircleinradius

Problem Statement

Let ΔABC\Delta ABC be a triangle with angle ∠CAB=60∘\angle CAB=60^{\circ}, let DD be the intersection point of the angle bisector at AA and the side BCBC, and let rB,rC,rr_B,r_C,r be the respective radii of the incircles of ABDABD, ADCADC, ABCABC. Let bb and cc be the lengths of sides ACAC and ABAB of the triangle. Prove that 1rB+1rC = 2⋅(1r+1b+1c) \frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)
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