Let 
,
, 
, and 
be four points on a circle,
and 
,
, 
, 
the orthocenters of triangles 
, etc. If, from the eight
points, four with different subscripts are chosen such
that three are from one set and the fourth from the other, these points
form an orthocentric system. There are eight
such systems, which are analogous to the six sets of orthocentric
systems obtained using the feet of the angle bisectors,
orthocenter, and polygon
vertices of a generic triangle.
On the other hand, if all the points are chosen from one set, or two from each set, with all different subscripts, the four points lie on a circle. There are four pairs of such circles, and eight points lie by fours on eight equal circles.
The Simson line of 
with regard to triangle 
is the same as that of
with regard to the triangle 
.
