Question 164
Let $B$ be any point on a circle $k_{1}$ and let $A$ be a point other than $B$ on the tangent to $k_{1}$ in $B$. Furthermore, let $C$ be a point outside $k_{1}$ with the property that the line $A C$ intersects the circle $k_{1}$ at two different points. Finally, let $k_{2}$ be the circle that touches the line $A C$ in $C$ and touches the circle $k_{1}$ in a point $D$, which is on the other side of $A C$ as $B$.
Prove that the center of the circle of triangle $BCD$ lies on the center of the circle of triangle $A B C$.
Prove that the center of the circle of triangle $BCD$ lies on the center of the circle of triangle $A B C$.