Supplementaire

Mineur includes these theorems in his work Cubiques Anallagmatiques.

Isotomic conjugation	Isogonic conjugation
When a triangle is inscribed in ABC is in perspective to ABC, it is again in perspective to the antimedial triangle, and the second center of perspective is the reciprocal [isotomic conjugate] of the first with respect to the antimedial triangle.	When a triangle is inscribed in ABC is in perspective to ABC, it is again in perspective to the antisupplementary triangle, and the second center of perspective is the inverse [isogonal conjugate] of the first with respect to the antisupplementary triangle.
When triangle A'B'C', inscribed in ABC, is in perspective with ABC, the conics BCB'C'G, CAC'A'G, ABA'B'G pass through the "anticomplementaire" of the reciprocal of the center of perspective.	When triangle A'B'C', inscribed in ABC, is in perspective with ABC, the conics BCB'C'Io, CAC'A'Io, ABA'B'Io pass through the "antisupplementaire" of the inverse of the center of perspective.

with examples

2.  If A'B'C' are the vertices of the orthic triangle, the conics BCB'C'I, CAC'A'I, ABA'B'I pass through the "antisupplementaire" of the center of the circumcircle. [This point is the reflection of the incenter in the circumcenter].

3.  If A'B'C' is the medial triangle, the conics BCB'C'I, CAC'A'I, ABA'B'I pass through the symmedian point of the excentral triangle [the Mittenpunkt].

4.  The circumference BIC passes through the antisupplementary of the point where the exterior bisectrices of A meet the circumcircle.

One says that a figure and the supplementary figure are homologous, the pole being I and the the axis being the anti-orthic axis [perspectrix of I].  Designate by M the point where the external bisector of A, meets the circumcircle, by M' the antisupplementary to M; by A' the point where MA meets BC. Then M' is the second point of intersection of MI with the arc BIC. Since Ia is the antisupplemtary to A, the property mentioned of supplementary figures shows that the line M'Ia goes through A'. Similarly BM, M'Ib and CM, M'Ic meet both on the antiorthic axis and on the sides of the triangle of reference.

Mineur clearly intends a parallel between the isotomic world and the isogonal world. Nowadays we would extend this parallel to all possible conjugations.

If a point has form :y: in trilinear coordinates then

:z+x: is its "supplementaire" and   :z+x-y: its antisupplementaire.

By their form these three points are obviously collinear with I_o(1:1:1). This eliminates the possibility that they have anything to do with the incentral and excentral triangles, for which corresponding points are not colinear.

Trilinears seem best to me using angles, so for angle functions here are some correlations that I have found by searching Kimberling's ETC.

This is part of my mining the ETC project, for which negative correlations are potentially as interesting as positive ones. I am interested in this operation because the classical geometers used it and because it is not mentioned in ETC, and thereby presumably by modern geometers – the general principle here being that finding out what is not said is often as interesting as what is said. I am also interested in points whose trilinears coordinates are half-angles of A, B, C because they do not relate well to the great mass of points and because they have unique extraversion properties.

Here is a table of some points (P) and their supplements (sP) and antisupplements (aP) collected by both Wilson Strothers and myself.

function	P	sP	aP	name of P
sin A	6	37	9	symmedian point
cos A	3	65	46	circumcenter
csc A	2	42	43	centroid
sec A	4	73	1745
tan A	19	48	610	Clawson pt
cot A	63	31	1707
sin A/2	266		164	258 on line
cos A/2	259		173
tan A/2	57	55	165
cot A/2	9	6	1743	note the relation to line 1
sec A/2	174		503
csc A/2	188		361

We have discovered much about these relationships

From Wilson:
aP = gP-Ceva conjugate of I [which I call the Cevian quotient]
gP = sp-cross conjugate of I.

Also, (P, sP, aP, I) is harmonic,
as is (P, aP, sP, X), where
X is the intersection of IP with the antiorthic axis [the tripolar of Io].

To these I add the tripolar of gP, to create symmetry between P and gP (the anallagmatic symmetry), obtaining the following figure.


In this picture we see the anallagmatic symmetry between P and gP. Notable is the P—gP line which contains gsP and gsgP. This is the Mineur line. The conjugate of this line is the Mineur circumconic which goes through P, gP, sP, and sgP.

gsP—sgP and sP—gsgP intersect on this conic at the fourth intersection of it and the circumconic with perspector Io

These last two properties seem universal among conjugations.

Wilson again:

Familiar operations can be derived from s,g and a( the inverse of s):

gsP = cevapoint of I, P,
sgP = crosspoint of I, P,
agP = P-ceva conjugate of I,
gaP = P-cross conjugate of I.

Algebraic symmetry suggests the inclusion of the cross-conjugate since we have inverse pairs gs,ag and ga,sg.

Conversely, if we know about Q-ceva conjugates and crosspoints:

aP = gP-ceva conjugate of I, so aP is the cevian quotient gP/I,
sP = crosspoint of gP and I.

Geometrically, we have collinearities including

I,P,sP,aP, and, replacing P by gP,
I,gP,sgP,agP,

and Steve's example
P, gsP, gP, gsgP

But there is further geometry which prompts two further operations.

We write

Cir(Q) for the circumconic with perspector Q,
Inc(Q) for the inconic with perspector Q.

gsP
= tripole of polar of P in Cir(I)
= tripole of polar of I in Cir(P)
sgP
= pole of tripolar of P in Inc(I)
= pole of tripolar of I in Inc(P)
agP
= pole of tripolar of P in Cir(I)
gaP
= tripole of polar of P in Inc(I)

It now appears that there are two "missing" operations:

xP = pole of tripolar of I in Cir(P) (the I-ceva conjugate of P)
yP = tripole of polar of I in Inc(P) (the I-cross conjugate of P)

I was surprised to find that these can be expressed using the basic operations a, s and g

xP = sgaP
yP = gsgagP

These make it clear that the operations have order 2.

Also, xP can be regarded as the crosspoint of I and aP, or as the supplement of the P-cross conjugate of I.

The operation zP = agsP also has order 2, but I can't see the geometry.
It is, however, ag(sP), the sP-ceva conjugate of I,
As such, it does make brief appearances in ETC (X1044,..,X1054).
It is also a(gsP), the antisupplement of the cevapoint of P and I.