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.

Dean, K. R. and van Lamoen, F. M. "Geometric Construction of Reciprocal Conjugates." Forum Geom.1, 115-120,
2001. http://forumgeom.fau.edu/FG2001volume1/FG200116index.html.van
Lamoen, F. M. "Bicentric Triangles." Nieuw Arch. Wisk.17,
363-372, 1999.