Question 44
Let $AB\Gamma$ be an acute angled triangle with $AB \lt A\Gamma \lt B\Gamma$. Its circumcircle is $c$ and let $\Delta, E$ be the midpoints of $AB$ and $A\Gamma$ respectively. We draw externally two semicircles with diameters $AB$ and $A\Gamma$, which intersect $E\Delta$ at $M$ and $N$ respectively. The lines $MB$ and $N\Gamma$ intersect the circumcircle at $T, \Sigma$ respectively. If the lines $MB$ and $N\Gamma$ intersect at $H$, prove that:
a) the point $H$ is on the circumcircle of the triangle $AMN$
b) the lines $AH$ and $T\Sigma$ intersect perpendicularly at the point $Z$ and $Z$ is the center of the circumcircle of the triangle $AMN$.
a) the point $H$ is on the circumcircle of the triangle $AMN$
b) the lines $AH$ and $T\Sigma$ intersect perpendicularly at the point $Z$ and $Z$ is the center of the circumcircle of the triangle $AMN$.