Geometry · Excircles · Tangent lengths · Constant perimeter · Family of tangent lines
A triangle \(ABC\) is given. Consider all lines which cut the side \(AC\) at a point \(M\) and the side \(BC\) at a point \(N\) in such a way that \(MN = AM + BN\). Prove that there is a circle \(k\) which touches every one of these lines.
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Serbian Republic Competition (Republicko takmicenje) 2002, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source