We start the article with already known theorem: Let a quadrilateral ABCD be given in the plane. Construct the orthocenters \(H_A\) , \(H_B\) , \(H_C\) and \(H_D\) of triangles BCD, ACD, ABD and ABC. Then the quadrilateral \(H_A H_B H_C H_D\) has the same area as the quadrilateral ABCD. In the paper, we give a synthetic proof of this theorem. In the framework of this proof we formulate another theorem, which is then generalized in various ways. Presented analytic proofs of these generalizations are based on Gröbner bases computation.
