Question 43
The circle $C$ has diameter $AB$. The tangent at $B$ is $T$. For each point $M$ (not equal to $A$) on $C$ there is a circle $C'$ which touches $T$ and touches $C$ at $M$. Find the point at which $C'$ touches $T$ and find the locus of the center of $C'$ as $M$ varies. Show that there is a circle orthogonal to all the circles $C'$.