Problem

Source: Kosovo National Olympiad 2025, Grade 12, Problem 3

Tags: geometry, similar triangles, national olympiad, Kosovo



Let $g_a$, $g_b$ and $g_c$ be the medians of a triangle $\triangle ABC$ erected from the vertices $A$, $B$ and $C$, respectively. Similarly, let $g_x$, $g_y$ and $g_z$ be the medians of an another triangle $\triangle XYZ$. Show that if $$g_a : g_b : g_c = g_x : g_y : g_z, $$then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.