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midpoint of median drawn from vertex A lies on axis of parabola

Source: 2023 South Russian Girls MO - Assara Seniors p7

March 1, 2024
analytic geometryconicsparabolageometry

Problem Statement

A parabola is drawn on the coordinate plane - the graph of a square trinomial. The vertices of triangle ABCABC lie on this parabola so that the bisector of angle BAC\angle BAC is parallel to the axis OxOx . Prove that the midpoint of the median drawn from vertex AA lies on the axis of the parabola.