Problem

Source: Cyprus 2022 Junior TST-2 Problem 3

Tags: geometry, isosceles, circumcircle



Let ABC be an acute-angled triangle, and let D,E and K be the midpoints of its sides AB,AC and BC respectively. Let O be the circumcentre of triangle ABC, and let M be the foot of the perpendicular from A on the line BC. From the midpoint P of OM we draw a line parallel to AM, which meets the lines DE and OA at the points T and Z respectively. Prove that: (a) the triangle DZE is isosceles (b) the area of the triangle DZE is given by the formula EDZE=BCOK8