[1] Lines {CC', BB', AA'} are parallel.

[2] The direction of these parallel lines is orthogonal to the direction of the Simson line of P.

[3] Triangle A'B'C' is always similar to the pedal triangle A

[4] The circumcircle of A'B'C' is centered at I and passes through P.

[5] The isogonal conjugate P* of P is on the line at infinity.

[6] Triangles ABC and A'B'C' are point perspective with respect to P*. Their line of perspectivity passes through the incenter I of ABC.

[1,5] follows by comparing angles, at A and B say (figure).

[2] follows similarly by comparing angles at P of the cyclic quadrilateral APP

[3,4] follow by noticing that, by their definition, I is on the medial lines of {PA', PB', PC'}. Then comparing the angles of A'B'C' with those formed at P.

[6] follows from an easy calculation with trilinears.

ax+by+cz=0, ({a,b,c} being the side-lengths of the triangle), whereas

ayz+bzx+cxy=0

is the equation of the circumcircle (see CircumcircleInTrilinears.html ).

In the file IsogonalOfParabola.html I discuss the special case of parabolas, resulting by taking the isogonal image of a tangent to the circumcircle.

IsogonalOfParabola.html

IsogonalGeneralized.html

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