Given a circle with center A and a point C, here are three related ways of constructing Pascal’s Limacon:
It is the isotomic (look it up) of the circle with pole C.
It is the conchoid (look it up) of a circle centered at A whose circumference passes through C.
It is an epitrochoid (look it up) formed by a circle of equal radius rolling around the original.
This app constitutes a visual proof of the above, and depends on the fact that the composition of reflections in two parallel lines is equivalent to a translation of twice the distance between the lines.