On a property of the orthocenter

A parallelogram, and more generally a convex quadrangle, defines four triangles, each consisting of two adjacent sides and a diagonal. Taking the orthocenters of these triangles, we form a new quadrangle and show it to be equiareal to the first one. We prove also an analogous property for parallelograms with vertices on the Euler lines of two such triangles having a common side of the parallelogram. In this case, we prove that there are, besides the quadruple of the orthocenters, three other quadruples defining parallelograms equiareal to the original one.