Select any three points, calculate orthocenter[1] and centroid[1]; those two points define line[1].

Use the other three points and calculate orthocenter[2] and centroid[2]; those two points define line[2].

The intersection of those two lines will give you the (h,k) coordinates.

The real difficulty of your problem is the generation of six coordinate points on a circle that average to (4/3,2/3).

If you supply the coordinates for the six points, the remainder is easy.