The following GoeGebra applets show the construction procedure for the Nine-point Circle in the Euclidean and Minkowskian spaces.
You can compare both procedures step by step clicking at the control arrows under each applet.
The Nine-point Circle joins the following points of a triangle:
-The midpoints of the three sides: Blue colour
-The feet of the three altitudes: Red colour
-The midpoints from the three vertexes to the orthocenter: Green colour
Drag the vertexes to change the shape of the triangle and the position of all the points.
EUCLIDEAN PLANE
MINKOWSKIAN PLANE
In this case, the euclidean circle is substituted bt its minkowskian counterpart:
the stCircle, which is an equilateral straight hyperbola.
Following the same procedure but using different tools (the "normal" Geogebra tools for the Euclidean case, and their corresponding "spacetime" tools for the Minkowskian case) renders completely equivalent geometric results.