Line TYZ associated with a triangle ABC
- Anton Zakharov
S - othocenter N - centroid O - incenter T - circumcenter U - 9-point center. points T, N, S, U belong to Euler line be definition. Point M can be constructed for points N and O by the method as defined here: https://math.stackexchange.com/questions/3748383/median-bisector Point Y can be constructed for points N and T applying the same method. Point Z can be constructed for points N and S in the exact same way. (Please note that in this case we are using the 6-gon inscribed into the 9-point circle of the triangle ABC. Continuations of the sides of the 6-gon intersect at 3 points that can be connected to the vertices of the original triangle ABC to get the point Z) Finally, points T, Y, Z belong to some unknown line, that apparently exists for any acute triangle ABC and point M doesn't lie on that line.