We prove that an elliptic triangle is congruent to its polar triangle if and only if six specific Wallace-Simson lines of the triangle are concurrent. (If a point projected onto a triangle has the three feet of its projections collinear, that line is called a Wallace-Simson line.) These six lines would be concurrent at the orthocenter. The six lines come from projecting a vertex of either triangle onto the given triangle. We describe how to construct such triangles and a dozen Wallace-Simson lines.

Author Bio

Nicholas Grabill saved this entire project in the summer of 2018 when, as a high school student, he rescued Wallace-Simson lines from the failed conjectures pile. Nicholas is an undergraduate student at Michigan who is interested in everything touching mathematics.

Kelsey Hall was a member of that summer 2018 team as well. She presented some of these ideas at an RHIT Undergraduate Math Conference. She finished her Masters at MSU in Biostatistics in May 2023 and is currently a doctoral student in the same program.

Morgan Nissen is a Canadian goalie who stepped into the elliptic Wallace-Simson line research in 2019. He found the I and Q points, which he named as these letters because math people have high IQ. He is currently a graduate student at Missouri S&T.

Noelle Kaminski did summer research in 2021. She found properties of elliptic triangles we need in this paper.

Jarrad Epkey's summer 2022 research project started with elliptic triangles congruent to polar triangles and all these other ideas from other summers fell into place. He has found that over-scheduling does not have an upper bound.