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For three points A,B,C in the plane, we define m(ABC) to be the smallest length of the three heights of the triangle ABC, where in the case A, B, C are collinear, we set m(ABC) = 0. Let A, B, C be given points in the plane. Prove that for any point X in the plane,

m(ABC) \leq m(ABX) + m(AXC) + m(XBC).

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