Professor Emeritus
Department of Mathematics
North Carolina State University at Raleigh
Raleigh, NC.
This manuscript: http://www.infiniteproduct.info/struclqd.htm
Curriculum Vitae: http://www.infiniteproduct.info/strublcv.htm

Let three lines in the Euclidean plane intersect at three points, A, B, and C. Let three other lines, passing through A, B, and C, concur at some point K, forming a complete quadrangle, where no vertex lies at the centroid of the triangle determined by the three other vertices. The latter three lines intersect the former three lines at three non-collinear (diagonal) points:

a=AK∩BC, b=BK∩AC, and c=CK∩AB

(occasionally at infinity). Prove that the three intersections α=bc∩BC, β=ac∩AC, and γ=ab∩AB (occasionally at infinity) are collinear.

In this figure, one sees triangle ABC, cevians Aa, Bb, Cc concurring at K, and intersections α, β, γ, as specified for the stated problem. This figure can be reinterpreted in three-space, where ABCK depicts a tetrahedron, and the fourth vertex K falls behind the front face ABC. In this interpretation, additional rays from α, β and γ depict a plane cutting through the tetrahedron and meeting plane ABC along the line passing through α, β and γ. The less visible configurations are obtained by movement of the fourth vertex K of the tetrahedron to other locations relative to ABC (multiple locations behind and in front of ABC). The proof in the reference then applies to the configurations produced by these other tetrahedra.

Amazingly, this collinearity, stemming from any complete quadrangle ABCK, appears to have been overlooked since the beginning. Focusing on a complete quadrangle, however, tends to obscure the relevant construction, unless one vertex, such as K, is viewed as having a special relationship internally or externally with respect to the remaining triangle, ABC.