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Found problems: 26

2016 Irish Math Olympiad, 10
Tags: euler circle , geometry
Let $AE$ be a diameter of the circumcircle of triangle $ABC$. Join $E$ to the orthocentre, $H$, of $\triangle ABC$ and extend $EH$ to meet the circle again at $D$. Prove that the nine point circle of $\triangle ABC$ passes through the midpoint of $HD$. Note. The nine point circle of a triangle is a circle that passes through the midpoints of the sides, the feet of the altitudes and the midpoints of the line segments that join the orthocentre to the vertices.