Prove BMXN is cyclic.

Question

Suppose $C_1$ and $C_2$ are circles such that {$,$}=$_{1}\cap _2$. We draw a secant $MN$ such that $\in _1$ and $\in _2$, and $A\in MN$. Show that if $X$ is the point of intersection of the tangents to $C_1$ and $C_2$ through $M$ and $N$ respectively, then the $$ quadrilateral is cyclic.

I have tried based on this exercise, but I cannot reach a conclusion, I would appreciate it very much if you could help me to solve it.

Answer 1

• Because of tangent-chord property we have $\angle XMA = \angle MBA = x$ and $\angle XNA = \angle NBA = y$
• Since $\angle MXN = 180-x-y$ and $\angle MBN = x+y$ we have $\angle MXN+\angle MBN = 180$ which means $MXNB$ is cyclic.