Prove that if the line through the centroid and the incenter is parallel to a side then the sides of the triangle form an arithmetic sequence.
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Suppose that (I=incenter, G=centroid). Let be the inradius, the height from the vertex A, the area of triangle and the semiperimeter. Then we have But and that means that b, a, c (or c, a, b) are in arithmetic progression.
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