Question 119
In the plane there is a triangle $ABC$ with circumcircle $\Gamma$; let $A'$ be the antipode of $A$ on $\Gamma$. Construct an equilateral triangle $BCD$ such that $A$ and $D$ lie on opposite sides of $BC$. Let the line through $A'$ perpendicular to $A'D$ meet the lines $AC$, $AB$ at $E$, $F$ respectively. On $EF$ as base, construct the isosceles triangle $ETF$ with base angles $30^\circ$, such that $A$ and $T$ lie on opposite sides of $EF$. Prove that $AT$ passes through the nine-point centre $N$ of triangle $ABC$.
Note: The nine-point circle of triangle $ABC$ is the circle passing through the nine points consisting of the midpoints of the three sides of triangle $ABC$, the feet of the three altitudes, and the midpoints of the three segments from the vertices to the orthocentre.
Note: The nine-point circle of triangle $ABC$ is the circle passing through the nine points consisting of the midpoints of the three sides of triangle $ABC$, the feet of the three altitudes, and the midpoints of the three segments from the vertices to the orthocentre.