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Medians of a triangle

Abstract

The medians of a triangle meet together at the barycenter of the triangle, $P$. To prove this I have worked out a representation of the triangle onto the trigonometric circle where it will determine three different real numbers $\alpha,\gamma, \beta$then $$ P = \left( \frac{\cos(\alpha)+ \cos(\gamma) + \cos(\beta)}{3}, \frac{\sin(\alpha)+ \sin(\gamma) + \sin(\beta) }{3} \right) $$ As byproduct I have determined the quotient set of the equivalence relation of triangles.