Let 
be a triangle perspective
to a reference triangle 
with perspector 
.
Let 
be the intersection of lines 
and 
, 
the intersection of 
and 
, and 
the intersection of 
and 
. Then 
is called the desmic mate of 
.
The desmic mate is perspective to both 
through a point 
and to 
through a point 
, and the perspectors are collinear. The twelve points are all perspectors of the
two quadrangles (
,
, 
) of which they are not a vertex and
can be viewed as the projection to the plane of a desmic
configuration.
The triangles 
,
and its desmic mate have a common
perspectrix. The trilinear
pole of this perspectrix lies on the line 
, more precisely it is the harmonic conjugate of 
w.r.t. 
and 
(van Lamoen 1999).
The pairs 
,
, 
and 
are pairs of the same isoconjugation.
The twelve points 
,
, 
, 
,
, 
, 
,
, 
, 
, 
, 
lie on an isocubic (Dean
and van Lamoen 2001) with pivot point 
.
