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a+t_a = b+t_b = k(a+b) , median sum identity

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1999 p3

February 20, 2020
medianSumratiogeometry

Problem Statement

Show that there exists a triangle ABCABC such that a≠ba \ne b and a+ta=b+tba+t_a = b+t_b, where ta,tbt_a,t_b are the medians corresponding to a,ba,b, respectively. Also prove that there exists a number kk such that every such triangle satisfies a+ta=b+tb=k(a+b)a+t_a = b+t_b = k(a+b). Finally, find all possible ratios a:ba : b in such triangles.
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