Properties of the Simson–Wallace locus applied on a skew quadrilateral

Let K,L,M be orthogonal projections of a point P onto the sides of a triangle ABC. Then the locus of P such that K,L,M are collinear, is the circumcircle of ABC. This theorem has several generalizations [4], [5], [10],[6], [7], [9]. A generalization of the Simson–Wallace theorem which is by [2] ascribed to J. D. Gergonne is as follows: Let K,L,M be orthogonal projections of a point P onto the sides of a triangle ABC. Then the locus of P such that the area of the triangle KLM is constant, is the circle through P which is concentric with the circumcircle of ABC. If we consider a tetrahedron ABCD instead of a triangle ABC then we can investigate the locus of points P ∈ E3 whose orthogonal projections onto the faces of ABCD are coplanar or form a tetrahedron of a constant volume. This was studied in [10], [6], [7], [9]. The generalization of Simson–Wallace theorem on skew quadrilaterals in the Euclidean 3D space is as follows [6], [8]: The locus of a point P whose orthogonal projections K,L,M,N onto the sides on a skew quadrilateral ABCD form a tetrahedron of a constant volume s is a cubic surface G. By searching for the locus and its properties we applied computer aided coordinate method based on Groebner bases computation and Wu–Ritt method using the software CoCoA [1] and the Epsilon library [11] working under Maple. The cubic surface G can be investigated from various points of view. In [8] reducibility of G with s = 0 was explored. The following conjecture was stated: The Simson–Wallace locus which is a cubic surface G is decomposable iff two pairs of sides a skew quadrilateral ABCD are of equal lengths. If for instance |AB|= |BC|= a and |CD|= |DA|= b, then in the case a = b the cubic G decomposes into a plane and a one-sheet hyperboloid, and if a = b we get three mutually orthogonal planes. In the talk further properties of G are studied. It is well known that the maximum number of lines of a general cubic surface is 27. There is a question how many lines do lie on the cubic G? It seems that the maximum number of lines lying on G is 15. This issue is also connected with the number of the so called