Locus of a point

[A dummy progression]

TL;DR Summary

How to find an angle equal to another when the lines forming the angles are movable

Could someone help me?

 

[mjc123]

Well, for a start, I think "square ABCD" is meant to imply that the points A, B, C, D go in that order round the perimeter, so C is diagonally opposite to A.

 

[PeroK]

It seems I am not the worst circle drawer in the world!

 

[mjc123]

Also, I think "circumscribed square" means the square is outside the circle (its sides are tangents to the circle). What you have drawn is the inscribed square.