I came across a maths problem, which I reduced to need the following result:
If $A,B$ are fixed points, $C,D$ are points not on line AB, and CD is a curve (in this case, a section of the circumference of the circle), and $\angle ACB>\angle ADB,$ is it necessarily true that $\forall \alpha$ such that$ \angle ACB<\alpha<\angle ADB$, $\exists$ a point $P$ on the curve such that $\angle APB=\alpha?$
I searched up on the internet with no result. Since the angle value is 'continous', I feel like this should be true, but I can't prove it (except the easy case where $CD$ is a line). Any idea on how to prove this in general (when $CD$ is a curve)?
Any help appreciated.