To find the formula for this combinaton of three reflections requires very little knowledge of complex numbers. All you have to do is to combine the complex maps as you combine functions.. The example shows the power of complex numbers as a tool for working with transformations in the plane.
In the problem Complex Rotations
you see another examle of using complex numbers for work with transformations.
The problems Rots and Refs
and Reflect Again
are examples of the use of matrices for work with transformations.
This problem is about combinations of reflections. The problem Reflect Again uses matrices to show that the combination of two reflections in intersecting mirror lines gives a rotation.