Maltitudes and Cyclic Quadrilaterals

Many results in quadrilateral geometry are traditionally stated for convex quadrilaterals. In this paper, we show that several of these results remain valid for self-intersecting quadrilaterals. In particular, we prove that, for any vertex, the maltitudes (midpoint altitudes) corresponding to two adjacent sides and the associated diagonal are concurrent; we call this point the malticenter of the vertex. This result holds uniformly for both simple and self-intersecting quadrilaterals. We further relate malticenters to classical centers (centroid, circumcenter, and orthocenter) via homotheties. Finally, we prove a conjecture of [1], previously observed for convex quadrilaterals, showing that \[ \operatorname{Area}(ABCD)=\operatorname{Area}(H_AH_BH_CH_D) \] holds for all quadrilaterals, where $H_A,H_B,H_C,H_D$ are the orthocenters of the triangles formed by the vertices of $ABCD$.