A triangle has the congruent isoscelizers point and equal parallelians point.
I'm pretty sure that any tetrahedron has many cutting planes that will cut an equilateral triangle. Is that much true -- that any three leg solid angle has equilateral triangle cutting planes?
If so, is there a point of concurrence for 4 of these cutting planes?
If so, is there a point where 4 congruent equilateral triangles get cut?
If not that, is there a point where 4 congruent triangles gets cut?