[1] Triangles PAB and PB'A' are similar.

[2] Line PD is orthogonal to A'B', D being the middle of AB.

[3] Quadrangle DOFP is a parallelogram, F being the middle of A'B' and O the center of c.

[4] Line A'B' is tangent to the fixed circle c' with center O and radius equal to the radius of c

[5] The locus of E is a limacon of Pascal (see Limacon.html for its definition).

Consider the circle c

[1] Follows by the equality of angles at {A,A'} and {B,B'}.

[2] Is a consequence of [1] by measuring the angles at P.

[3] By [1] applied to PA'B', PF is orthogonal to AB. It follows that quadrangle DOFP has parallel opposite sides.

[4] Is a consequence of [3].

[5] Follows from the definition of the limacon (see Limacon.html ) and the previous properties.

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