The P-line is easier in this model, since a circle orthogonal to p has its center on p. For typical A and B, this is the construction of the P-line AB:
For C and D when line CD is perpendicular to p:
The desired P-circle is orthogonal to any P-line through E, including EE', so the E-center of the circle must be on EE'. Another P-line through E is EF; the tangent E-line to the supporting circle of EF at F also contains the center of the P-circle, so this constructs the center as the intersection of two lines. Then the P-circle is the E-circle with this intersection as center.
Here is another construction that works also when EF is perpendicular to the E-line p. Construct the P-line n through E orthogonal to EE', as shown. Intersect with the E-line EE' the radical axis of F and the supporting circle of n to get the center of the P-circle.
The construction of this P-line is the construction of a circle with center on p so that G is inverted to H. This means that the center O of the circle is also on E-line GH. The radius of the circle is the geometric mean of OG OH and can be constructed as the length of a tangent from O to any circle through G and H.
Of course on such example of a circle through G and H is the P-line GH:
Special Case when GH is parallel to p. The P-perpendicular bisector is the E-perpendicular bisector.
