The Two Incenters of the Arbitrary Convex Quadrilateral

TL;DR

This paper introduces two special incenters within an arbitrary convex quadrilateral, relates their associated circle radii to the quadrilateral's area and perimeter, and explores special cases where these incenters coincide or have equal radii.

Contribution

It defines two incenters on the Newton line of a convex quadrilateral and establishes a new formula linking the quadrilateral's area, perimeter, and the harmonic mean of the radii of tangent circles.

Findings

The area equals half the product of perimeter and the harmonic mean of the radii.

Conditions for the incenters to coincide or have equal radii are characterized.

New geometric relationships between incenters and quadrilateral properties are derived.

Abstract

For an arbitrary convex quadrilateral $ABCD$ with area ${\cal A}$ and perimeter $p$, we define two points $I_1, I_2$ on its Newton line that serve as incenters. These points are the centers of two circles with radii $r_1, r_2$ that are tangent to opposite sides of $ABCD$. We then prove that ${\cal A}=pr/2$, where $r$ is the harmonic mean of $r_1$ and $r_2$. We also investigate the special cases with $I_1\equiv I_2$ and/or $r_1=r_2$.