Kulkarni-Ahona Corollary for the Newton-Gauss Theorem

We give a new corollary approach to the Newton-Gauss Line, showing collinearity using coordinate geometry and an algebraic expansion method in the Cartesian coordinate plane. Building upon principles from the Newton-Gauss line and properties of a complete quadrilateral, we prove that the midpoints of the three diagonals of a complete quadrilateral are collinear. Notable methods of proving said theorem include proof by similarity, similar triangles, comparing areas, ratios, and proof by other geometric approaches. We formulate a corollary approach to the Newton-Gauss Line using coordinate geometry by mapping the Newton-Gauss diagram on the Cartesian coordinate plane with one quadrilateral vertex at the origin. We then assign coordinates to the other vertices, find the midpoints of the diagonals, and show the collinearity of the three key points, considering the shared point at the intersection of both lines. Furthermore, we build upon ideas of a complete quadrilateral through a right-angle quadrilateral on the base of the Cartesian coordinate plane, satisfying the definitional ideas of the Newton-Gauss line and its geometric makeup.