Parabolic polygons

TL;DR

This paper proves a geometric property of parabolas and parabolic quadrilaterals, establishing conditions for the existence of an inscribed circle tangent to sides, with an elementary synthetic proof and interesting corollaries.

Contribution

It provides a new characterization of parabolic quadrilaterals with inscribed circles based on orthogonal diagonals, with an elementary proof and novel corollaries.

Findings

A circle tangent to sides exists iff diagonals are orthogonal.

Elementary synthetic proof of the main theorem.

Several elegant corollaries derived from the main result.

Abstract

Main Theorem. Two parabols have four common points. There exists a circle tangent to the sides of the obtained parabolic quadrilateral if and only if the diagonals of this quadrilateral are orthogonal. The proof of the Main Theorem is elementary and purely synthetic. It is based on the following lemma. Assume that a parabola is tangent to a circle at points A and B. A point P of the plane lyes on the parabola if and only if the distance from the point P to the line AB equals to the length of the tangent from P to the circle. We present some beautiful elementary corollaries of the Main Theorem.