The Two Incenters of the Arbitrary Convex Quadrilateral

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 $r_1=r_2$.