Question 1
In the plane, a circle $(O)$ and two distinct fixed points $B$ and $C$ on it are given, with $BC$ not a diameter. $A$ varies on $(O)$, with $A$ different from $B$ and $C$. Let $D, K$ and $J$ be the midpoints of $BC, CA$ and $AB$, respectively. Let $E, M$ and $N$ be the perpendicular projections of $A, B$ and $C$ onto the lines $BC, DJ$ and $DK$, respectively.
Prove that the tangents at $M$ and $N$ to the circumcircle of triangle $EMN$ meet at a fixed point $T$ as $A$ varies.
Prove that the tangents at $M$ and $N$ to the circumcircle of triangle $EMN$ meet at a fixed point $T$ as $A$ varies.