Nikolaos Dergiades and Dimitris M. Christodoulou, The two incenters of an arbitrary convex quadrilateral,
Forum Geometricorum, 17 (2017) 245--254.

Abstract. For an arbitrary convex quadrilateral ABCD with area 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 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.

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