Find a point C on an infinite line AB which, when connecting two other points M and N, would form congruent angles

Question

On an infinite line $AB$, find a point $C$ such that the rays $CM$ and $CN$ connecting $C$ with two given points $M$ and $N$ situated on the same side of $AB$ would form congruent angles with the rays $CA$ and $CB$ respectively.

I think $C$ will be the vertical axis of symmetry - but I'm not sure how to construct this line (and therefore the point $C$ on the segment $AB$) with a compass and ruler. Can somebody point me in the right direction?

Answer 1

This is an important problem in optics, where the line $\overleftrightarrow{AB}$ is a mirror, point $M$ is an eye and point $N$ is an object. The point $C$ is then the point on the mirror between the eye and the object's image.

Just drop a perpendicular from point $N$ to line $\overleftrightarrow{AB}$ and find the point $N'$ that is the same distance from the line as $N$ is. Point $C$ is then the intersection of line $\overleftrightarrow{AB}$ and line segment $\overline{MN'}$.

Point $N'$ is the image of point $N$ in the mirror.

Answer 2

The two angles are the same when the perpendicular to $AB$ through $C$ bisects the $\widehat{MCN}$ angle.

Let $M',N'$ be the symmetric of $M,N$ with respect to the $AB$-line.

Can you prove that when we take $C$ as $AB\cap MN'$ that is exactly the case?

Hint: $\widehat{BCN}=\widehat{ACN'}$.