Let Ab be the intersection between AC and the perpendicular to BC from B. Let Ac the intersection between AB and the perpendicular to BC from C. Define cyclically Bc, Ba, Ca, Cb. Then:
1) The six points Ab, Ac, Bc, Ba, Ca, Cb lie in a conic.
2) A', B', C' being the midpoints of AbAc, BaBc, CaCb, the lines AA', BB', CC' concur.
3) A”, B”, C” being the midpoints of BaCa, CbAb, AcBc, the lines AA”, BB”, CC” concur at the De Longchamps point L.
4) A*, B*, C* being the midpoints of BcCb, CaAc, AbBa, the lines AA*, BB*, CC* concur at the symmedian K.