Question 51
Let $k_{1}$ and $k_{2}$ be circles with centers $O_{1}$ and $O_{2}$, $O_{1}O_{2}=25$, and radii $R_{1}=4$ and $R_{2}=16$, respectively. Consider a circle $k$ such that $k_{1}$ is internally tangent to $k$ at a point $A$, and $k_{2}$ is externally tangent to $k$ at a point $B$.
a) Prove that the segment $AB$ passes through a constant point (i.e., independent on $k$).
b) The line $O_{1}O_{2}$ intersects $k_{1}$ and $k_{2}$ at points $P$ and $Q$, respectively, such that $O_{1}$ lies on the segment $PQ$ and $O_{2}$ does not. Prove that the points $P, A, Q$ and $B$ are concyclic.
c) Find the minimum possible length of the segment $AB$ (when $k$ varies).
a) Prove that the segment $AB$ passes through a constant point (i.e., independent on $k$).
b) The line $O_{1}O_{2}$ intersects $k_{1}$ and $k_{2}$ at points $P$ and $Q$, respectively, such that $O_{1}$ lies on the segment $PQ$ and $O_{2}$ does not. Prove that the points $P, A, Q$ and $B$ are concyclic.
c) Find the minimum possible length of the segment $AB$ (when $k$ varies).