Half a year ago, I have noted that certain affine constructions
based on a triangle ABC and a point P provide concurrences;
some of these concurrences are well-known, others not, but the
possibly new thing is that the points of concurrence coincide.
So I begin with a definition:

The isotomcomplement Q of a point P with relation to a
triangle ABC is the complement of the isotomic
conjugate of P.

Here, "complement" means the same as "inferior", "subordinate"
or "medial image", i. e. the complement of a point with
respect to a triangle is the image of this point in the
homothety centered at the centroid of this triangle and having
factor -1/2.

Now, I have proven using barycentrics:

THEOREM 1: If A'B'C' is the cevian triangle of P, and A1, B1,
C1 are the midpoints of its sides B'C', C'A', A'B', then
the lines AA1, BB1, CC1 concur at the isotomcomplement Q
of P with respect to ABC.

The concurrence (without the fact that the point of
concurrence is Q) is easy to prove (it is a special case of
cevian nests). A corollary of Theorem 1 is:

THEOREM 2: The isotomcomplement Q of P is the center of
the cevian inconic of P, i. e. the conic touching BC,
CA, AB at the cevian feet A', B', C' of P.

(I had also treated isotomcomplements on 1 Jul 2001 in a
geometry-college message, but now I am much further.)

A different affine construction leading to the
isotomcomplement is the following one:

THEOREM 3: If A'B'C' is the cevian triangle of P, and Ma,
Mb, Mc, Ma', Mb', Mc' are the midpoints of BC, CA, AB,
AA', BB', CC', then the lines MaMa', MbMb', McMc'
concur at the isotomcomplement Q of P.

Note that Theorem 3 can be easily proven synthetically.

Here is a very different construction I am proud of:

THEOREM 4: If parallels to the sides B'C', C'A', A'B' of
the cevian triangle of P are drawn through A, B, C, then
the triangle they enclose is perspective to ABC, the
perspector being the isotomcomplement Q of P.
Moreover, this triangle is the anticevian triangle of Q.

I. e. the cevian triangle of a point and the anticevian
triangle of its isotomcomplement are homothetic.

Finally, we note that

THEOREM 5: If a point P has barycentrics P(x:y:z), then its
isotomcomplement Q has barycentrics
Q( x(y+z) : y(z+x) : z(x+y) ).

THEOREM 6: If a point P has trilinears P(x':y':z'), then its
isotomcomplement Q has trilinears
Q( x'(by'+cz') : y'(cz'+ax') : z'(ax'+by') ).

We conclude by some special cases of isotomcomplements:

THEOREM 7: The following table of points P and their
isotomcomplements Q can be proven:

point P isotomcomplement Q

(a) centroid centroid
(b) orthocenter symmedian point
(c) Gergonne point incenter
(d) Nagel point Mitten point
(e) incenter "simplest unnamed center",
X(37)
(f) circumcenter X(216) - is this the only
geometric property of it?
(g) symmedian point Brocard midpoint, X(39)

Note that from Theorem 7(b) and Theorem 3, we conclude the
(not very simple) fact that the lines joining the midpoints
of the sides of ABC with the midpoints of the altitudes
concur at the symmedian point.

But now we move to the real hit of the theory (see
Hyacinthos #6380 about cyclocevian conjugates):

THEOREM 8: If two points are cyclocevian conjugates, then
their isotomcomplements are isogonal conjugates.

This can be proven synthetically by Theorem 1. Note that
a restatement of Theorem 8 gives a direct construction of
the cyclocevian conjugates (see also Hyacinthos #462):

THEOREM 9: The cyclocevian conjugate of a point is the
isotomic conjugate
of the anticomplement
of the isogonal conjugate
of the complement
of the isotomic conjugate
of the point.

This is not only a nice description of the cyclocevian
conjugate, but it also makes possible an elementary
derivation of its barycentric coordinates. In fact, the
coordinate formula given in Paul Yiu's Introduction to
Triangle Geometry,

http://www.math.fau.edu/yiu/GeometryNotes020402.ps

can be easily deduced from Theorem 9.

So far the main parts of the isotomcomplement theory. Like
one could imagine, there are many other affine constructions
leading to isotomcomplements. Here is a conjecture (I am
sure that there is a simple barycentric proof, but would
like a synthetic one):

If the cevian triangle A'B'C' of a point P is dilated
with center the isotomcomplement Q of P, then the
image is a triangle perspective with ABC.

Since the isotomcomplement of the Gergonne point is the
incenter (Theorem 7(c)), the Kariya theorem is a special
case of this conjecture.

Sincerely,
Darij Grinberg