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Area of trapezium is six times that of triangle [Cyprus IMO TST 2018]

Source: Cyprus IMO TST 2018, Problem 2

July 1, 2018

geometrytrapezoid

Problem Statement

Consider a trapezium

AB \Gamma \Delta

, where

A\Delta \parallel B\Gamma

and

\measuredangle A = 120^{\circ}

. Let

E

be the midpoint of

AB

and let

O_1

and

O_2

be the circumcenters of triangles

AE \Delta

and

BE\Gamma

, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle

O_1 E O_2