The triangle congruence axiom that led to ellipse collinearity

This paper aims to prove the assumption that the center of the circle containing two vertices of a complete quadrilateral, along with the center of the conic section formed by the two diagonal points of the quadrilateral (with the third diagonal point being fixed), are collinear with the midpoint of the segment defined by the intersection of their common tangent lines. The common tangent lines are drawn from the two remaining vertices of the complete quadrilateral. Furthermore, a general theorem can be deduced from this proof, asserting that when a conic section is considered instead of a circle, the centers of the two conic sections and the midpoint of the section formed by the intersections of their common tangent lines are collinear. The assumptions were discovered and proved using the dynamic mathematics software GeoGebra, which offers many visual tools such as the Show Trace or Locus tool, the CAS View, and the 3D calculator. These tools were employed to provide analytic proof. Consequently, a correlation has been identified that converts a projective correspondence into an affine property.