Question 2
Let $ABC$ be a triangle in the plane, and let $D$ be the intersection of segment $AB$ with the internal bisector of $\angle ACB$. A variable circle $(O)$ always passes through $C,D$ and is not tangent to either line $BC$ or $CA$. It meets $BC,CA$ again at $M,N$, respectively.
(a) Prove that there exists a circle $(S)$ tangent to lines $DM,DN$ at $M,N$, respectively.
(b) Circle $(S)$ meets lines $BC,AC$ at $P,Q$, respectively. Prove that the lengths $MP,NQ$ are constant as $(O)$ varies.
(a) Prove that there exists a circle $(S)$ tangent to lines $DM,DN$ at $M,N$, respectively.
(b) Circle $(S)$ meets lines $BC,AC$ at $P,Q$, respectively. Prove that the lengths $MP,NQ$ are constant as $(O)$ varies.