We outline a synthetic proof strategy based on radical axes and power of points. We show that the tangency condition is equivalent to a relation among the powers of H with respect to three circles: Ω, Γ, and the circle with diameter CD.

We provide a detailed analysis of the configuration under inversion centered at one intersection point. We derive explicit relationships among the images of the key points and reduce the tangency condition to a simple property involving cross ratios and orthogonal circles.

We use inversion about A to transform the configuration into a simpler one where the two original circles become lines. We then reduce the tangency condition to a property of the inverted figure.

We propose an inversion-based method to prove that the line through the orthocenter of a certain triangle parallel to a certain line is tangent to the circumcircle of another triangle formed by intersection points of two circles. The inversion simplifies the configuration and reduces the problem to a more manageable statement.

We provide strong numerical evidence that the line through H parallel to AP is tangent to the circumcircle of BEF, using extensive random testing across admissible parameters.

We show that the orthocenter H of triangle PMN lies on the same vertical line as the circumcenter P of triangle ACD.

We prove that in the configuration of two intersecting circles with centers M and N, the line through the orthocenter of triangle PMN parallel to line AP is tangent to the circumcircle of triangle BEF, using analytic geometry and symbolic verification.

We study a configuration involving two intersecting circles and prove that the orthocenter of a certain triangle lies on the perpendicular bisector of a segment formed by the intersections of the line of centers with the circles. This is a step towards proving the full statement that a line parallel to a certain line through this orthocenter is tangent to the circumcircle of a triangle formed by the other intersection points.