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.
Epkey, Jarrad S.; Nissen, Morgan; Kaminski, Noelle K.; Hall, Kelsey R.; and Grabill, Nicholas
"Elliptic triangles which are congruent to their polar triangles,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 24:
2, Article 9.
Available at: https://scholar.rose-hulman.edu/rhumj/vol24/iss2/9