Make a triangle with vertices A, B, and C.
Find the Midpoints of the 3 sides, call them L, M and N.
Find the feet of altitude
for the three vertices and call the point at which the intersect the opposite
sides D, E, and F
The point at which all the altitudes intersect is called the orthocenter and lets label that H.
Now find the midpoints of the AH, BH, and CH. Call these points P, Q and, R
Now you can see all the points that are constructed: L, M, N D, E, F, P, Q, R.
Notice you can pass a circle through all these points
How do we find the centre of this nine point circle?
draw the circumcircle of the original triangle, A, B, and C
find the centre of the circumcircle, call it CC
draw a line connecting the orthocenter, H and the circumcentre CC
find the midpoint of the line segment CC and H and this is U. U is the centre of the nine point circle