Reciprocal conjugates constructed from a triangle
- Dear all,
Keith Dean and I have been discussing some constructions for reciprocal
conjugacies (terminology?) lately. That is conjugacies mapping
x:y:z --> u/x : v/y : w/z.
Here is a short description of a very surprising one (for me at least):
Let A'B'C' be the triangle with coordinates:
Let X be the point x:y:z.
Let A" = XA'/\BC, B" and C" likewise.
Then let A_1 = B'C"/\B"C', B_1 and C_1 likewise.
Then A_1B_1C_1 is perspective to ABC through the point:
Uu/x : Vv/y : Ww/z
This is particularly interesting because this maps the perspector P of
A'B'C' to the perspector of its desmic mate. Not only the perspector,
the vertices of A'B'C' itself are also mapped to the vertices of the
desmic mate. In fact this reciprocal conjugacy is of course the
conjugacy of Barry Wolk's desmic cubic:
x(u+U)(wW yy - vV zz) + y (v+V)(uU zz - wW xx) + z(w+W) (vV xx - uU yy)
Realizing this, it is not difficult to find a construction of the
reciprocal conjugacy extending one (proper) given pair of points.