Invariant Triangle Structure

(Created 10/22/05) These pages give a classification of triangle points based on considerations of symmetry, and note important relationships. A constructive notation for triangle centers is introduced. Conic are introduced and lightly covered.

Natural points and lines

The idea is to set up points, lines, and conics in a way that their relationships exhibit the maximum possible symmetry. Consider three points in ABC, G, the centroid, an arbitrary point P and P' = tP, its isotomic conjugate. This structure is invariant in that any affine transformation will preserve the stated relationships.
The three points have tripolars lines. (G's is the line at infinity). In addition the three points can be connected to create the three lines G—P, G—P', and P—P'. Hence there are six natural lines in this system.
Actually in affine invariant situations, the dual works as well as the tripolar and we will use that more often, writing the dual to point P as ~P. Since the dual of the line G—P is the intersection of the dual of P with the line at infinity, we have the notational formula

~ (G—P) = ~ P●∞, (1)

where the dot indicates intersection.
This system of 3 points and six lines is affine invariant and invariant under the interchange of a point with its conjugate. The ancients called this interchange  "anallagmatic symmetry." Note that the line P—P' is, by itself, anallagmatically symmetric. Mineur made a big deal of this line, so I call it the Mineur line.
We have 3 points G, P, P' and 6 lines ∞, ~P, ~P', G—P, G—P', P—P'.
There are 15 intersections of these 6 lines. Three are the points G, P, P' and 7 other points in the finite triangle plane. There are 5 intersections at infinity:

(2)

Points on the Steiner ellipse

The isotomic conjugates of the 5 points at infinity are on the Steiner elllipse. These points obey the following relationships. The isotomic operation is indicated by the prefix "t".

t((G–P)●∞), t((~P)●∞), and G are collinear.
t((G–P')●∞),   t((~P')●∞), and G are collinear.
t((G–P')●∞),  t((~P)●∞), and P are collinear.
t((G–P)●∞),  t((~P')●∞), and P' are collinear.

What is nice about these structures is the interrelations of these points and lines.

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Figure: This shows the 3 natural pointsin red and their 6 natural lines in blue. The Seiner circumellipse and the Mineur conic are shown.The red lines connect points the isotomics of the infinite points of the blue lines. The Mineur point is at the lower left.

The Mineur point, the Mineur line, the Mineur conic

The point where the duals of P and P' meet is the Mineur point and is notated as  ~P●~P'. This point is analagmatically symmetric and is the tripole of the Mineur line. It is the perspector of the Mineur conic, which has wonderful properties, and which is the isotomic conjugate of the Mineur line.

The Affine Parallelogram

Three affine invariant operations are t, the isotomic conjugate; m, the medial operation the is P defined wrt the medial triangle; and d, its inverse, called the "dilated operation" in the triangle book, which give P defined to the dilated or anti-medial triangle. The following picture gives these operations applied to the original three points.

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Generate lines from existing points; generate new points from the affine operations and as intersection of lines.
There is unexpected structure.
Theorem 1: The points P,  tmP, tmP’, and P’ are colinear.
Proof: Using middle coordinates only, tmP ~ :(l+m)(m+n): = :m(l+m+n) + nl:, which shows that tmP is colinear with tP and P. Similarly tmP’ ~ nl(m+n)(m+l) = nl(mn+nl+lm) + m·nml, which shows that it is on the same line in a different place.
Theorem 2: The lines P—dP’, mP—P’ are parallel as are the lines P’—dP and mP’—P.  
Proof: We take the difference of the (normalized) points to get the vector direction: P – mP’ = P’– dP ~  l2m – l2 n – ln2 + mn2, where only the middle coordinate is shown. Of course this is the infinite point in its direction, a point that will show up later in out system. Similarly P – dP’ = P’–mP ~  m(l2 – lm – mn + n2).
These lines form a parallelogram as shown in the figure (the green lines). I call this the “affine parallelogram” of P (also of tP). It is structurally very important.

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Figure: The affine parallelogram, showing its relation to triangle points and to the Mineur conic.

Classification of points and lines

This system of generating points and lines (and eventually conics and cubics) is an automaton that generates points of higher and higher complexity. The list has the same points at Clark Kimberling's ETC if the generating point P is the incenter.
The list is generated by only a few operations: the dual operation, the isotomic conjugate, and the medial operation and its inverse. These operations do not commute. These operations preserve strength (i.e., they never take strong to weak or vice versa).
Surprisingly (very surprisingly, imo) no other conjugation is needed. Isogonic conjugates, for example, are naturally generated by the system. See the example section on the orthocenter.
There is a new constructive notation for all points.
The system is recursive in that the structure of points derived from a point P can be immediately applied to the derived points.

32 Points

Continuting to generate points by the above system, we get these 32 points by going several levels deep. The incidence column shows the index of lines the point is on. The lines are shown in the following table.

index name incidence coordinates
0 G
3
4
6
9
12
1
1
1
1 P
3
5
7
10
13
19
29
l
m
n
2 tP
4
5
8
11
14
15
24
m n
l n
l m
3 (~P•∞)
0
1
6
7
8
16
20
25
30
35
37
m-n
-l+n
l-m
4 (~tP•∞)
0
2
9
10
11
17
21
26
31
36
38
-l (m-n)
m (l-n)
-(l-m) n
5 (~tP•~P)
1
2
12
13
14
18
22
27
32
-l (m-n) (m+n)
m (l-n) (l+n)
-(l-m) (l+m) n
6 mP
3
15
16
17
18
23
33
op
m+n
l+n
l+m
7 mtP
4
19
20
21
22
23
28
op
l (m+n)
m (l+n)
(l+m) n
8 dP
3
24
25
26
27
28
34
op
-l+m+n
l-m+n
l+m-n
9 dtP
4
29
30
31
32
33
34
op
l m+l n-m n
l m-l n+m n
-l m+l n+m n
10 ((P—G)•∞)
0
3
-2 l+m+n
l-2 m+n
l+m-2 n
11 ((P—G)•~P)
1
3
-l m + m^2 - l n + n^2
l^2 - l m - m n + n^2
l^2 + m^2 - l n - m n
12 ((P—G)•~tP)
2
3
-l (l m+l n-2 m n)
-m (l m-2 l n+m n)
n (2 l m-l n-m n)
13 ((tP—G)•∞)
0
4
l m+l n-2 m n
l m-2 l n+m n
-2 l m+l n+m n
14 ((tP—G)•~P)
1
4
m (2 l-m-n) n
-l n (l-2 m+n)
-l m (l+m-2 n)
15 ((tP—G)•~tP)
2
4
l (l m^2 - m^2 n + l n^2 - m n^2)
m (l^2 m - l^2 n - l n^2 + m n^2)
-n (l^2 m + l m^2 - l^2 n - m^2 n)
16 ((tP—P)•∞)
0
5
23
34
(m + n) (l^2 - m n)
-(l + n) (-m^2 + l n)
-(l + m) (l m - n^2)
17 ((tP—P)•~P)
1
5
m n (2 l^2 - m^2 - n^2)
-l n (l^2 - 2 m^2 + n^2)
-l m (l^2 + m^2 - 2 n^2)
18 ((tP—P)•~tP)
2
5
l (l^2 m^2 + l^2 n^2 - 2 m^2 n^2)
m (l^2 m^2 - 2 l^2 n^2 + m^2 n^2)
-n (2 l^2 m^2 - l^2 n^2 - m^2 n^2)
19 t(~P•∞)
op
-(l-m) (l-n)
(l-m) (m-n)
-(l-n) (m-n)
20 t(~tP•∞)
op
-(l-m) m (l-n) n
l (l-m) (m-n) n
-l m (l-n) (m-n)
21 t(~tP•~P)
op
-(l-m) m (l+m) (l-n) n (l+n)
l (l-m) (l+m) (m-n) n (m+n)
-l m (l-n) (m-n) (l+n) (m+n)
22 tmP
5
op
(l+m) (l+n)
(l+m) (m+n)
(l+n) (m+n)
23 tmtP
5
op
m (l+m) n (l+n)
l (l+m) n (m+n)
l m (l+n) (m+n)
24 tdP
24
op
-(l+m-n) (l-m+n)
(l-m-n) (l+m-n)
(l-m-n) (l-m+n)
25 tdtP
29
op
(l m-l n-m n) (l m-l n+m n)
(l m-l n-m n) (l m+l n-m n)
-(l m+l n-m n) (l m-l n+m n)
26 m(~tP•~P)
12
35
36
op
(m - n) (l^2 + m n)
-(l - n) (m^2 + l n)
-(-l + m) (l m + n^2)
27 mmP
3
op
2 l+m+n
l+2 m+n
l+m+2 n
28 mmtP
4
op
l m+l n+2 m n
l m+2 l n+m n
2 l m+l n+m n
29 d(~tP•~P)
12
37
38
op
(l+m) (m-n) (l+n)
-(l+m) (l-n) (m+n)
(l-m) (l+n) (m+n)
30 ddP
3
op
3 l-m-n
-l+3 m-n
-l-m+3 n
31 ddtP
4
op
-l m-l n+3 m n
-l m+3 l n-m n
3 l m-l n-m n

Point 11 is a very interesting point and deserves a few comments. It is related to the Steiner inverse of P and leads to the following notational equation

Steiner inverse of P = d ((G—P) ● ~ P) = (G—P) ● ~ mP (3)

The second equality requires some explaining, but not now.

39 Lines

This shows 39 lines defined by symmetry consideration. The incidence column gives the index of points from the above chard that are incident with the line. The coodinates are baycentric line coordinates.

index name incidence coordinates
0 ∞
3
4
10
13
16
1
1
1
1 ~P
3
5
11
14
17
l
m
n
2 ~tP
4
5
12
15
18
m n
l n
l m
3 (P—G)
0
1
6
8
10
11
12
27
30
m-n
-l+n
l-m
4 (tP—G)
0
2
7
9
13
14
15
28
31
-l (m-n)
m (l-n)
-(l-m) n
5 (tP—P)
1
2
16
17
18
22
23
-l (m-n) (m+n)
m (l-n) (l+n)
-(l-m) (l+m) n
6 ((~P•∞)—G)
0
3
-2 l+m+n
l-2 m+n
l+m-2 n
7 ((~P•∞)—P)
1
3
-l m + m^2 - l n + n^2
l^2 - l m - m n + n^2
l^2 + m^2 - l n - m n
8 ((~P•∞)—tP)
2
3
-l (l m+l n-2 m n)
-m (l m-2 l n+m n)
n (2 l m-l n-m n)
9 ((~tP•∞)—G)
0
4
l m+l n-2 m n
l m-2 l n+m n
-2 l m+l n+m n
10 ((~tP•∞)—P)
1
4
m (2 l-m-n) n
-l n (l-2 m+n)
-l m (l+m-2 n)
11 ((~tP•∞)—tP)
2
4
l (l m^2 - m^2 n + l n^2 - m n^2)
m (l^2 m - l^2 n - l n^2 + m n^2)
-n (l^2 m + l m^2 - l^2 n - m^2 n)
12 ((~tP•~P)—G)
0
5
26
29
(m + n) (l^2 - m n)
-(l + n) (-m^2 + l n)
-(l + m) (l m - n^2)
13 ((~tP•~P)—P)
1
5
m n (2 l^2 - m^2 - n^2)
-l n (l^2 - 2 m^2 + n^2)
-l m (l^2 + m^2 - 2 n^2)
14 ((~tP•~P)—tP)
2
5
l (l^2 m^2 + l^2 n^2 - 2 m^2 n^2)
m (l^2 m^2 - 2 l^2 n^2 + m^2 n^2)
-n (2 l^2 m^2 - l^2 n^2 - m^2 n^2)
15 (mP—tP)
2
6
l^2 (m - n)
-m^2 (l - n)
(l - m) n^2
16 (mP—(~P•∞))
3
6
l^2 - m n
m^2 - l n
-l m + n^2
17 (mP—(~tP•∞))
4
6
-l^2 m - l m^2 - l^2 n + 2 l m n + m^2 n - l n^2 + m n^2
-l^2 m - l m^2 + l^2 n + 2 l m n - m^2 n + l n^2 - m n^2
l^2 m + l m^2 - l^2 n + 2 l m n - m^2 n - l n^2 - m n^2
18 (mP—(~tP•~P))
5
6
-(l+m) (l+n) (l m+l n-2 m n)
-(l+m) (m+n) (l m-2 l n+m n)
(l+n) (m+n) (2 l m-l n-m n)
19 (mtP—P)
1
7
-m (m-n) n
l (l-n) n
-l (l-m) m
20 (mtP—(~P•∞))
3
7
l^2 m - l m^2 + l^2 n + 2 l m n - m^2 n - l n^2 - m n^2
-l^2 m + l m^2 - l^2 n + 2 l m n + m^2 n - l n^2 - m n^2
-l^2 m - l m^2 - l^2 n + 2 l m n - m^2 n + l n^2 + m n^2
21 (mtP—(~tP•∞))
4
7
-m n (l^2 - m n)
l n (-m^2 + l n)
l m (l m - n^2)
22 (mtP—(~tP•~P))
5
7
-m (l+m) (2 l-m-n) n (l+n)
l (l+m) n (l-2 m+n) (m+n)
l m (l+m-2 n) (l+n) (m+n)
23 (mtP—mP)
6
7
16
(l+m) (m-n) (l+n)
-(l+m) (l-n) (m+n)
(l-m) (l+n) (m+n)
24 (dP—tP)
2
8
24
l (l-m-n) (m-n)
m (l-n) (l-m+n)
-(l-m) (l+m-n) n
25 (dP—(~P•∞))
3
8
2 l^2 - l m + m^2 - l n - 2 m n + n^2
l^2 - l m + 2 m^2 - 2 l n - m n + n^2
l^2 - 2 l m + m^2 - l n - m n + 2 n^2
26 (dP—(~tP•∞))
4
8
-l (l m + m^2 + l n - 4 m n + n^2)
-m (l^2 + l m - 4 l n + m n + n^2)
-n (l^2 - 4 l m + m^2 + l n + m n)
27 (dP—(~tP•~P))
5
8
-l^3 m - l^2 m^2 - l^3 n + 2 l^2 m n + l m^2 n - m^3 n - l^2 n^2 + l m n^2 + 2 m^2 n^2 - m n^3
-l^2 m^2 - l m^3 - l^3 n + l^2 m n + 2 l m^2 n - m^3 n + 2 l^2 n^2 + l m n^2 - m^2 n^2 - l n^3
-l^3 m + 2 l^2 m^2 - l m^3 + l^2 m n + l m^2 n - l^2 n^2 + 2 l m n^2 - m^2 n^2 - l n^3 - m n^3
28 (dP—mtP)
7
8
-(m - n) (l^2 + l m + l n + 2 m n)
(l - n) (l m + m^2 + 2 l n + m n)
-(l - m) (2 l m + l n + m n + n^2)
29 (dtP—P)
1
9
25
(m-n) (l m+l n-m n)
-(l-n) (l m-l n+m n)
-(l-m) (l m-l n-m n)
30 (dtP—(~P•∞))
3
9
-l m^2 + 4 l m n - m^2 n - l n^2 - m n^2
-l^2 m - l^2 n + 4 l m n - l n^2 - m n^2
-l^2 m - l m^2 - l^2 n + 4 l m n - m^2 n
31 (dtP—(~tP•∞))
4
9
l^2 m^2 - 2 l^2 m n - l m^2 n + l^2 n^2 - l m n^2 + 2 m^2 n^2
l^2 m^2 - l^2 m n - 2 l m^2 n + 2 l^2 n^2 - l m n^2 + m^2 n^2
2 l^2 m^2 - l^2 m n - l m^2 n + l^2 n^2 - 2 l m n^2 + m^2 n^2
32 (dtP—(~tP•~P))
5
9
l^3 m^2 - 2 l^3 m n - l^2 m^2 n + l m^3 n + l^3 n^2 - l^2 m n^2 - 2 l m^2 n^2 + m^3 n^2 + l m n^3 + m^2 n^3
l^2 m^3 + l^3 m n - l^2 m^2 n - 2 l m^3 n + l^3 n^2 - 2 l^2 m n^2 - l m^2 n^2 + m^3 n^2 + l^2 n^3 + l m n^3
l^3 m^2 + l^2 m^3 + l^3 m n - 2 l^2 m^2 n + l m^3 n - l^2 m n^2 - l m^2 n^2 + l^2 n^3 - 2 l m n^3 + m^2 n^3
33 (dtP—mP)
6
9
(m - n) (2 l^2 + l m + l n + m n)
-(l - n) (l m + 2 m^2 + l n + m n)
(l - m) (l m + l n + m n + 2 n^2)
34 (dtP—dP)
8
9
16
(m - n) (l^2 + m n)
-(l - n) (m^2 + l n)
(l - m) (l m + n^2)
35 ~dP
3
26
dual
-l+m+n
l-m+n
l+m-n
36 ~dtP
4
26
dual
l m+l n-m n
l m-l n+m n
-l m+l n+m n
37 ~mP
3
29
dual
m+n
l+n
l+m
38 ~mtP
4
29
dual
l (m+n)
m (l+n)
(l+m) n

The affine operations

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Examples

This chart shows the points defined by the three operations t, m, and d for 4 choices of P. The indices are not the same as the above chard since it contains only points defined by affine operations, but no duals or tripoles. The numbers next to point definitions are the Kimberling X numbers. This shows the extent for which my new system and the Kimberling system overlap. They overlap most for the orthocenter (desmon) column.

[Graphics:HTMLFiles/index_83.gif]

The points not represented in the Kimberling system (such as 15 and 21) are actually quite interesting as perhaps I will explain someday but not now.

Example 1: The incenter

This picture shows the affine parallelogram for P = incenter. A large number of points are coherently organized. The Mineur line is the Gergonne-Nagel line. The Mineur conic is the Feuerback hyperbola which contains Go, No, Io, and the Mittenpunkt Mo as well as H.

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Example 2: The orthocenter

This picture shows the affine parallelogram for P = orthocenter. A large number of points are coherently organized. The second diagonal is surprisingly the Euler line of the Brocard image triangle, also known as the first Brocard triangle. The i operator indicates the point with respect to the Brocard image triangle.
The Mineur line is the H—tH line. The Mineur conic is the Jerabek hyperbola of which the Mineur point is the perspector of this hyperbola.

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Conics

The point  t((~P)●∞) has a truly amazing property that is the reason for this discussion:
Suppose I "project" a point from the steiner ellipse to another circumconic, say the one with P as perspector. Let Q be a point on the Steiner ellipse, then this point is the barycentric product PQ. (This is a foundation of conjugation, I think).

The amazing thing is that the line Q—PQ goes through t((~P)●%).

So, for example, we can project a point Q from the Steiner ellipse to the circumcircle by drawing a line from the Steiner point through P the intersection with the circumcircle being the point.

I figure this result must by known by the people that study conjugations, but it is my highly symmetric way of introducing the subject.

In[364]:=

pinf = Cross[p, centroid]

Out[364]=

{m - n, -l + n, l - m}

In[456]:=

tpinf = l m n Cross[tp, centroid]/l/m/n//Simplify

Out[456]=

{l (-m + n), m (l - n), (-l + m) n}

In[380]:=

gpinf = Cross[pinf, centroid]

Out[380]=

{-2 l + m + n, l - 2 m + n, l + m - 2 n}

In[458]:=

gtpinf = l m  n Cross[tpinf, centroid]/l/m/n//Together

Out[458]=

{l m + l n - 2 m n, l m - 2 l n + m n, -2 l m + l n + m n}

In[381]:=

Cross[iso[gpinf], centroid]//Together

Out[381]=

{(3 (m - n))/((l + m - 2 n) (l - 2 m + n)), (3 (l - n))/((l + m - 2 n) (2 l - m - n)), -(3 (l - m))/((2 l - m - n) (l - 2 m + n))}

In[379]:=

Cross[iso[pinf], centroid]//Together

Out[379]=

{(-2 l + m + n)/((l - m) (l - n)), (-l + 2 m - n)/((l - m) (m - n)), (-l - m + 2 n)/((m - n) (-l + n))}

Which is the same line as the previous.

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