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Found problems: 1

2017 Adygea Teachers' Geometry Olympiad, 2
Tags: exradii , geometry , exradius , excircle
It turned out for some triangle with sides $a, b$ and $c$, that a circle of radius $r = \frac{a+b+c}{2}$ touches side $c$ and extensions of sides $a$ and $b$. Prove that a circle of radius $ \frac{a+c-b}{2}$ is tangent to $a$ and the extensions of $b$ and $c$.