I am thinking about, is there analogy beetween 2d and solid geometry about Trillium theorm. (But im intrest in only acute or right triangles)
Trillium theorem in 2d geometry holds:
Let ABC be triangle, let I be inscribed circle center in triangle triangle $ABC, midpoint on arc AB (on circle with points ABC) which doesn't include point C.
Therefore |AI| = |BI| = |ID|
(and points, A, I, D are on one line)
Is there any generalization of it in solid geomery like,
Let ABCD be a tetrahedron, let I be the inscribed sphere center in tetrahedron ABC, let line IA, meet the sphere which include points A, B, C and D at point E (diffrent than A), do it means, that |EC| =|BE| = |DE| = |IE|?