Natural points, natural lines, natural conics—invariant triangle structure

(Updated 2/22/06) 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. Circumconics are introduced and covered.

Natural points and lines

We wish to set up points, lines, and conics in a way that their relationships exhibit the maximum symmetry and greatest possible naturalness.

Consider three points in the plane of the reference triangle ABC: G, the centroid, an arbitrary point P with P' = tP, its isotomic conjugate. This three point structure is invariant in that any affine transformation will preserve these relationships and symmetric in that switching each point with its conjugate leaves the whole unchanged.

Each point has a corresponding line, its tripolar. (G's is the line at infinity). In addition the points can be connected to create the three lines G—P, G—P', and P—P', making six natural lines.

The dual

In affine invariant situations, the dual, being the tripolar of the conjugate, works as well as the tripolar. We write the dual of point P as ~P. This is a proper dual.
P = (l:m:n) has dual lx+my+nz = 0 and vice versa.

To be a proper dual the dual of a line between two points must be the intersection of the duals of the two points.

~(A—B) = ~A• ~B

The dual of the line G—P is the intersection of the dual of P with the line at infinity, so we have the notational formula

where the dot denotes intersection. This notational system is constructive; the same notation both identifies a point and gives some of its constructions.

This system of 3 points and six lines is both 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 we call it the Mineur line.

In our system we now have 3 points G, P, P' and 6 lines ∞, ~P, ~P', G—P, G—P', P—P'. We will proceed to generate points, lines, and conics in the most natural way, always maintaining invariance and symmetry.

Points at infinity

There are 15 intersections of these 6 lines. Three are the points G, P, P'. 7 other points lie in the finite triangle plane. These are listed below as part of a large list. There are 5 intersections at infinity, which are the endpoints of the five finite lines. We often call points at infinity "directions." The "direction" of a line is its point of intersection with infinity.

Points on the Steiner ellipse

Each direction corresponds to a point on the Steiner ellipse, since the conjugate of the line at infinity is the Steiner ellipse. The five infinite points obey the relationships in the following chart. The isotomic operation is indicated by the prefix "t".

This shows, among other things, that the directions of ~P and G—P map to antipodal points on the Steiner ellipse.

The following picture shows the three original points (red), five of the six original lines (blue), the five points on the Steiner ellipse (light blue), and the four lines defined above (red). The blue hyperbola is the Mineur conic defined below.

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

~P and G—P

The point P naturally generates these two lines. As we have seen above, the infinite points on ~P and G—P go to opposite points on the Steiner ellipse. The intersection of these two lines is also interesting. It is mP^2-, which is the medial of the Steiner inverse of P. This leads to the following notational formula:

The second equality requires some explaining, but not now. Hopefully I will have time to explain the mathematics of this notational system.

This prominent intersection probably explains why Steiner inverses (Conway's "2-" notation) come into triangle geometry so often and so fundamentally.

A nice result is that the dual of any point on G—P is parallel to ~P.
The dual of mP2– is parallel to ~P through P. The dual of the infinite point on ~P is G—P.
Proof: If line Lx+My+Nz goes through G, then L+M+N = 0. Since P = (l:m:n) is on the line, lL+mM+nN = 0. Any line parallel to the dual has the form lx+my+nz + λ(x+y+z) = (l+λ)x+(m+λ)y+(n+λ)z.   (l+λ,m+λ,n+λ) is on the original line if  (l+λ)L+(m+λ)M+(n+λ)N = lL+mM+nN + λ(L+M+N) = 0, as it does.

The duals of points on ~P go through P. The dual of mP^2– is parallel to ~P through P. The dual of the infinite point on ~P is G—P. Here is an expanded version of this section.

The Mineur point, the Mineur line, the Mineur conic

This section describes a point, line, and conic of high symmetry.

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 dual 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 point t((P'—P)?∞ ), the isotomic of the intersection of the Mineur line with the line at infinity, is the fourth intersection of the Mineur conic with the Steiner ellipse. The Mineur conic is shown in most pictures.

Invariant operations

Three affine invariant operations are t, the isotomic conjugate; m, the medial operation (P defined wrt the medial triangle), and d, its inverse, called the "dilated operation" in The Triangle Book, which gives P defined to the dilated or anti-medial triangle. The following picture gives these operations applied to the original three points (red). Under conjugation the three points go to eachother, so there are 4 new points. All lie on the original 3 lines.

The five lines; five circumconics

The isotomic conjugate of a line is a circumconic (unless the line goes through a vertex) whose perspector is the dual of the line. The P—P' line creates the Mineur conic through P, P', mP, and mP'. The isotomics of the G—P and G—P' lines are each conics with perspector at infinity. The isotomics of ~P and ~P' have perspectors P' and P respectively.

The following chart shows these 6 circumconics including the Steiner ellipse. The last column shows that most of these conics are familiar ones when P is familiar.

conic	perspector	from line	contains A, B, C, and	if P = H	□
Steiner ellipse	G	∞	4th	Steiner ellipse	□
Mineur conic	~P●~P'	P—P'	P, P', mP, mP', tQ	Jerabek hyperbola	□
P circumconic	P	~P	t(~P●∞), t(~P●~P')	H-circumconic	□
P' circumconic	P'	~P'	t(~P'●∞), t(~P●~P')	D(= tH) circumconic	□
□	~P'●∞	G—P'		□	□
□	~P●∞	G—P		Kiepert hyperbola	□

The P circumconic

The point  t((~P)●∞) on the Steiner ellipse has an amazing property:
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 P·Q, a projective transformation. The amazing thing is that the line Q—P·Q always goes through t((~P)●∞).

As an example, we can project a point Q from the Steiner ellipse to the circumcircle (the K-circumconic) by drawing a line from the Steiner point t((~K)●∞) through Q, the intersection with the circumcircle being the point.

Conjugate Operations

As an interesting aside we can algebraically conjugate the above operations, an operation familiar from group theory. If P and Q are operations, then PQP^-1, is an operation of the same type as Q. We give two examples. Conjugates of points operations seem to transform points related to conics.These conjugates are different from isotomic conjugates.

dtm is a conjugate of t. This operation takes the perspector of a circumconic to its center.
tmt is a conjugate of m. It takes a point on a conic generated from G—P or G—P' to another point on the same conic.

The dual of a circumconic is an inconic

The dual of a point on a circumconic of perpector P is a line tangent to an inconic of perspector tP. Of course, being a dual, it works both ways. The dual of a point on an inconic is a line tangent to a circumconic.

Corresponding points

There is a nice formula relating a point on a circumconic to the corresponding point (the point of tangency) on its dual.

Let the circumconic, perspector P = (l:m:n) have equation l/x + m/y + n/z = 0
Its corresponding inconic has equation √ lx + √ my + √ nz = 0
If Q = ( : y1 : ) is on this conic then ... + m/y1 + .... = 0. The dual of Q is x1 x + y1 y + z1 z = 0
The point ( : m/y1^2 : ) is manifestly on this line as well as the inconic. So this must be the point of tangency.

Hence the transformation ( x : y : z ) -> ( l/x1^2 : m/y^2 : n/z1^2) takes a point on a circumconic to a point on its dual inconic.

The natural conics using P = K and H

The dual of a point on a circumconic of perpector P is a line tangent to an inconic of perspector tP. Of course, being a dual, it works both ways. The dual of a point on an inconic is a line tangent to a circumconic.

Here is a picture showing how points lines and conics are naturally developed by the same affine invariant ideas. This shows the srong points and conics. Go here for more.

The Affine Parallelogram

We generate new lines from existing points and new points from the affine operations and as intersection of lines.

We quickly find 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.

Note: The analogue of this is true for all conjugations.

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 next figure (the green lines). I call this the “affine parallelogram” of P (also of P' = tP).

About the Affine parallelogram

The Mineur line P—P' is one diagonal of this parallelogram. The line connecting G to the center of the Mineur conic is the other. The midpoint between P and P' is its center. The points mP and mP' are midpoints of two of the edges.

The second diagonal contains G in a ratio 1:2. Letting Q be a vertex of the parallelogram on this diagonal, then mQ is the midpoint and dQ the other vertex. The center of the Mineur conic is on this line and may be taken as its defining point (along with G).

Triangles of Centers

The following triangles are congruent: ΔQmPmP'; ΔmQmP'mP; ΔmPP'mQ; ΔmP'mQP. There are also similar triangles. If the starting point P = H, the orthocenter his leads to, among other things, John Conway's triangles of centers. Examples of this structure for particular choices of P are given below.

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

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 largely 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, in my opinion) no other conjugation is needed. Isogonic conjugates, for example, are naturally generated by the system. See the example section on the orthocenter. Of course the isogonal conjugate, or other conjugations, will be more convenient in cirtain situations, but they are not strictly necessary.

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 two levels deep. The incidence column shows the index of lines the point is on. The lines are given in the following table.

Of course this table can be continued ad infinitum.

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
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
16	((tP—P)•∞)	0 5 23 34
17	((tP—P)•~P)	1 5
18	((tP—P)•~tP)	2 5
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
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. It is the intersection of G—P and ~P and has been mentioned before.

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
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
12	((~tP•~P)—G)	0 5 26 29
13	((~tP•~P)—P)	1 5
14	((~tP•~P)—tP)	2 5
15	(mP—tP)	2 6
16	(mP—(~P•∞))	3 6
17	(mP—(~tP•∞))	4 6
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
21	(mtP—(~tP•∞))	4 7
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
26	(dP—(~tP•∞))	4 8
27	(dP—(~tP•~P))	5 8
28	(dP—mtP)	7 8
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
31	(dtP—(~tP•∞))	4 9
32	(dtP—(~tP•~P))	5 9
33	(dtP—mP)	6 9
34	(dtP—dP)	8 9 16
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

This pattern shows using just the three operations, m, t, and d. It does not include intersctions, duals, or points at infinity. It does generate many known points given well known starting points.

Examples

The chart below shows a chart of 32 points for specific choices of the defining point using the above automaton, adding five of the more prominent intersections.. The number next to point definitions is the Kimberling X number. This shows the extent for which my new system and the Kimberling system overlap. They overlap most for the orthocenter (desmon) column.

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. Actually "someday" is past, the explanation is here.

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, and has Fo as its center. Its perspector is : b(c-a) sb : .

The next picture shows many of the same points with the three natural conics of the system: the Steiner ellipse, the Io-circumconic, and the Mineur conic, the Feuerbach hyperbola (red). Points such as wS and wT are weak analogues of the Steiner and Tarry points.

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.

The next picture shows another view, this time with the three natural conics, the Steiner ellipse, the circumcircle, and the Mineur conic which is the Jerabed hyperbola. The triangles of centers are shown. The affine parallelogram is light yellow. Inconics are dashed.

A more complete picture

This figure shows yet more of the points. It adds two more conics, those with perspectors P and P' with centers mtdP and mtdP' and intersection the isotomic Mineur point.

Which is the same line as the previous.

Created by Mathematica  (February 24, 2006)