HYACINTHOS 7984

In the new ETC by Clark Kimberling, the points
X(1601)-X(1634) are constructed using a transformation
Kimberling names "TCC-perspector".

Here are some notes on this transformation. I call it
"Steinbart point".

"Steinbart" because of the name given by Oliver Funck
in his work avaliable online at

http://www.uni-duisburg.de/SCHULEN/STG/Wettbewerbe/Jufo2.html

See also my note "Variations of the Steinbart Theorem"
on my website,

http://de.geocities.com/darij_grinberg/

In this note, I show an Extended Steinbart Theorem.
In one direction (i. e., without its converse), it
states:

- The circumcevian triangle of any point P with
respect to a triangle ABC is perspective with the
tangential triangle of ABC.
- The circumanticevian triangle of any point P with
respect to a triangle ABC is perspective with the
tangential triangle of ABC.

With "circumanticevian triangle" of P, I mean the
triangle whose vertices are the second intersections
of the anticevians of P with the circumcircle of ABC.
The anticevians of P are the harmonic lines of the
lines AP, BP, CP with respect to the sides of the
triangle (or, in other words, the sidelines of the
anticevian triangle of P).

Now, it turns out that the perspector for the
circumcevian triangle and the perspector for the
circumanticevian triangle are the same!

I call this perspector the "Steinbart point of P".
Kimberling calls it "TCC-perspector of P".

Now, we see that the Steinbart point of P coincides
with the Steinbart points of the triharmonic points of
P.

(The triharmonic points of a point are the vertices of
its anticevian triangle. The triharmonic points are
better known as "harmonic conjugates" with respect to
the triangle.)

Now, the Steinbart theorem (that every circumcevian
triangle is perspective with the tangential triangle)
can be generalized projectively: Take a circumconic
instead of the circumcircle and the tangential
triangle with respect to that conic instead of the
tangential triangle. Any "circumconiccevian" triangle
[whose vertices are the intersections of cevians with
the circumconic] is perspective to the tangential
triangle with respect to this conic.

Here we find a nice analogy:

The perspector of the cevian triangle of a point P
and the anticevian triangle of a point Q (i. e., the
P-Ceva conjugate of Q) has trilinears

( u ( - u/x + v/y + w/z )
: v ( u/x - v/y + w/z )
: w ( u/x + v/y - w/z ) ),

where (x:y:z) are the trilinears of P and (u:v:w) are
those of Q.

If Q is the symmedian point of triangle ABC, we have
u:v:w = a:b:c, and the perspector, i. e. the perspector
of the cevian triangle of P and the tangential
triangle of ABC, has trilinears

( a ( - a/x + b/y + c/z )
: b ( a/x - b/y + c/z )
: c ( a/x + b/y - c/z ) ).

The perspector of the CIRCUMcevian triangle of P and
the tangential triangle of ABC (this perspector is
the Steinbart point of P) has trilinears

( a ( - a²/x² + b²/y² + c²/z² )
: b ( a²/x² - b²/y² + c²/z² )
: c ( a²/x² + b²/y² - c²/z² ) ).

An immediate analogy. Now, if we have an arbitrary Q
with trilinears (u:v:w), take the circumconic touching
touching the sides of the anticevian triangle of Q at
A, B, C, then the "circumconiccevian" triangle of P
with respect to this conic is perspective to the
anticevian triangle of Q (which is the tangential
triangle of ABC with respect to this conic), and the
perspector has trilinears

( u ( - u²/x² + v²/y² + w²/z² )
: v ( u²/x² - v²/y² + w²/z² )
: w ( u²/x² + v²/y² - w²/z² ) ).

I would call this the P-Steinbart conjugate of Q.
"Conjugate" not because it is a real conjuagacy (it
is not), but for its analogy with the Ceva conjugacy.

Probably, Steinbart conjugates (or, better,
pseudo-conjugates) will be as prolific in defining new
points as their special case with Q = symmedian point.

Darij Grinberg