An intersection circle problem

Jan 2020
2
1
Novi Sad
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.
 
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Feb 2010
738
162
I'll bite. Maybe 540 degrees?
 
Jan 2020
2
1
Novi Sad
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
Dec 2006
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The answer seems to be 180°, but I haven't found a proof yet.
 
Feb 2010
738
162
I misread the problem. Skipjack is right ... 180 degrees.

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

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

These should be enough to help you.
(How do you do an arc symbol in latex? I just used \widehat.)
 

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skipjack

Forum Staff
Dec 2006
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$\begin{align*}\angle MAN + \angle MBN &= \angle MAN + (360^\circ - \angle MBA - \angle NBA) \\
&= \angle MAN + (180^\circ - \angle MBA) + (180^\circ - \angle NBA) \\
&= \angle MAN + \angle NMA + \angle MNA \\
&= 180^\circ \end{align*}$