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

A. whose centroids coincide with the centroid of the primary triangle.

B. whose incenters coincide with the incenter of the primary triangle.

C. whose orthocenters coincide with the orthocenter of the acute primary triangle.

D. whose circumcenters coincide with the circumcenter of the primary triangle.

E. whose Gergonne points coincide with the Gergonne point of the primary triangle.

A. Accurately drawn graphs provide overwhelming evidence that the desired non-concurrent cevians pass through the median vertices of homeothetic subtriangles. I trust that some solvers will provide rigorous justifications for this synthetic construction.

B. Given an initial cevian, the two desired companion cevians can be obtained (in turn) by doubling their angles relative to rays through the common incenter.

C. Given an initial cevian, the two desired companion cevians can be obtained by a proper selection of their intersection with the perpendicular to the initial cevian, which passes through the common orthocenter.

D. Given a reduced circle, centered at the primary (and common) circumcenter, the three non-concurrent cevians can be obtained by a proper selection of the three intersections with the given reduced circle.

E. This is a major challenge, tractable by a graphical iteration procedure. Along an initial cevian, one selects two prospective intersection points (on opposite sides of the primary Gergonne point), which then determine the third prospective intersection point (using the other cevians). Constructing rays from these three prospective intersection points, passing through the primary (common) Gergonne point, produces three intersections along the edges of the prospective subtriangle. One then modifies the positions of the two prospective intersections along the intial cevian, so as to best accommodate circular arc intersections at these two intersections with the three edge intersections. Iterations of this construction procedure will produce the three desired non-concurrent cevians as limits. Perhaps some solvers will supply a better solution.