$A$ and $B$ are two points on same side of straight line $L$, such that $AB$ is not parallel to $L$. Thus line $AB$, when produced, cuts $L$ at point $P$. $AB$ subtends two maximal angles on $L$, one on either side of $P$. Let these angles be $\alpha$ and $\beta$, and the two points be $M$ and $N$. Distance between $M, N$ is $c$.
I realised that to subtend maximal angles, $M$ must be the point of tangency of the circle passing through $A,B$ and tangent to $L$. Reasons are somewhat obvious, and I'm not stating them. Same goes for $N$.
Using simple geometry, I found $MBA$, $BMP$ are equal. Similarly $NAB$, $BNP$ are also equal.
Also, I got $MP=PN=c/2$. (I proved this by coordinate geometry)
Using these informations, I tried solving the problem by trigonometry, assuming angles $MAB=x, NAB=y$. But the process gets too complicated, and I cannot eliminate x, y from the final expressions. Any suggestions/hints or solutions would be very helpful. Thank you!