# An intersection circle problem

#### Vedast

The circles k1 and k2 intersect at points A and B. The common tangent touches them at points M and N. Calculate the sum of the convex angles ∠ MAN and ∠ MBN.

idontknow

#### mrtwhs

I'll bite. Maybe 540 degrees?

#### Vedast

Yeah, but I need an explanation :/ Also it can't be 540 because the angles must be less than 180 degrees, so maximum 360.

#### skipjack

Forum Staff
The answer seems to be 180°, but I haven't found a proof yet.

#### mrtwhs

I misread the problem. Skipjack is right ... 180 degrees.

$\angle NMA = \dfrac{1}{2} \widehat{AM} = \angle MBA$

$\angle MNA = \dfrac{1}{2} \widehat{AN} = \angle NBA$

\begin{align*}\angle MAN + \angle MBN &= \angle MAN + (360^\circ - \angle MBA - \angle NBA) \\ &= \angle MAN + (180^\circ - \angle MBA) + (180^\circ - \angle NBA) \\ &= \angle MAN + \angle NMA + \angle MNA \\ &= 180^\circ \end{align*}