Geometry · Incircle · Touch chord · Tangent lengths · Cyclic quadrilaterals · Median to the hypotenuse · Midline · Areas with a common base
The circle inscribed in a triangle \(ABC\) touches the sides \(AB\) and \(AC\) at the points \(M\) and \(N\) respectively. Let \(P\) be the point in which the bisector of the angle \(ABC\) meets the line \(MN\). Prove that the area of the triangle \(ABC\) is twice the area of the triangle \(ABP\).
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Serbian Republic Competition (Republicko takmicenje) 2003, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source