Problem

Source: Romanian National Olympiad 2024 - Grade 10 - Problem 2

Tags: geometry, circumcircle, pentagon, inscriptible, Centroid



We consider the inscriptible pentagon $ABCDE$ in which $AB=BC=CD$ and the centroid of the pentagon coincides with the circumcenter. Prove that the pentagon $ABCDE$ is regular. The centroid of a pentagon is the point in the plane of the pentagon whose position vector is equal to the average of the position vectors of the vertices.