I found a conjectures of generalization of the Parry circle theorem a more than year ago, but I no a found solution for the conjectures. I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a proof?
Parry circle theorem: Let $ABC$ be a triangle, the triangle centroid, the first and the second isodynamic points, the far-out point, the focus of the Kiepert parabola, the Parry point and two points in Kimberling centers $X(352)$ and $X(353)$ lie on a circle.
A generalization of Parry theorem: Let a rectangular circumhyperbola of ABC, let $L$ be the isogonal conjugate line of the hyperbola. The tangent line to the hyperbola at the orthoencenter meets $L$ at point $K$. The line through $K$ and center of the hyperbola meets the hyperbola at $F_+$, $F_-$. Let $ I_+$, $I_-$, $G$ be the isogonal conjugate of $F_+, F_-$ and $K$ respectively. Let $F$ be the inverse point of $G$ with respect to the circumcircle of $ABC$. Then five points $I_+$, $I_-$, $G$,X(110), $F$ lie on a circle. Furthermore $K$ lie on the Jerabek hyperbola.
When the hyperbolar is the Kiepert hyperbola the conjecture be comes Parry circle theorem. (see 1, 2)
1-https://groups.yahoo.com/neo/groups/AdvancedPlaneGeometry/conversations/messages/2255 2-http://tube.geogebra.org/m/1645609