nonEuclidean structure of triangle points

by Steve Sigur

Abstract: Examination of orbits of a group introduced by Clark Kimberling leads to curves of the form yⁿ ± zx = 0. The mathematics of these orbits leads to an understanding of the distribution of important triangle points.

Dedicated to the compilers of triangle points and curves.

It is easy to think that triangle geometry has too many points and too few justifications for them. A few years ago Clark Kimberling published a paper on central points and lines in the triangle [1]. He commented that there is a group operation on points which can be nicely expressed in terms of homogeneous coordinates. Paul Yiu later gave a construction [2] for an affine invariant version of this operation. I was excited when I heard of this group operation as it seemed a way of giving structure to the seemingly unmanageable growth in triangle centers. But after a little thought, Kimberling’s group did not seem to help much.

The structure of this group is easy to understand. The group objects, as originally defined, are centers, defined as points whose coordinates (and implicitly their methods of construction) are symmetric enough. The group operation is that corresponding coordinates multiply. Using barycentric coordinates the centroid, whose coordinates are (1:1:1), is the identity. Let P = :m:, then P² = P·P = :m²: and Pⁿ = :mⁿ: for n an integer form an orbit of this group, and any center in the plane of the triangle can be written as a direct sum of various orbits. (Note: “:b:” is shorthand for ( a : b : c), giving only the middle coordinate.) Eight years ago this orbital structure did not seem to help much. On any orbit, one quickly ran into the coordinates of points one had never heard of. So either this group was not very relevant or the wisdom of relevant points in triangle geometry was incorrect. It was easy to decide that the group was unhelpful. But one of the great features of this age of geometric discovery is that we know many more points and their properties than we did then, so perhaps this group is worth another look.

Notation

AP is the Cevian trace of P; i.e., the intersection of Cevian AP with edge BC and a vertex of the Cevian triangle of P. AP is the A-harmonic associate of P; i.e., the harmonic conjugate of P with respect to A and AP and a vertex of the pre-Cevian triangle of P. We follow John Conway’s notation for triangle points, as given in reference 3. “t” and “g”” represent the isotomic and isogonal conjugates. “p” = gt and “r” = tg are projective operations that map the Steiner ellipse to the circumcircle and vice versa. Details about particular points are given in an appendix. Coordinates are delimited by commas if they are normalized and by colons if not.Extraversion indices are used for points such as the incenter that have more than one instance so that the central incenter is Io while its version in the A-direction is Ia.

The plane of the triangle contains points, such as the orthocenter, that move long distances as the shape of the triangle varies. There are also points whose barycentric coordinates are not functions of the edges or angles such as (1,2,3). These points, and the structures composed from them, keep the same relative position to each other as the shape of the triangle changes. I call these points the affine background; some are shown in Fig. 1. I often think of the triangle plane as having an affine structure related to each point, being particular lines, points, conics, and cubics, which non-affine points pick up as they move across the points of the affine background. For example, (2:3:4) is the incenter for a certain triangle and is the perpector of an inscribed ellipse. If the shape of the triangle changes (an affine transformation) this ellipse still exits and (2:3:4) is still its perspector, but this point is no longer the incenter, which has moved and taken over the ellipse associated with some other point of the affine background. This point of view will be explained in more detail in a future article.

If P ~ (x₁:y₁:z₁) and Q ~ (x₂:y₂:z₂) are two points, P•Q ~ ( x₁x₂ : y₁y₂ : z₁z₂ ) is the group operation. The centroid G ~ (1:1:1) is the identity and tP, the isotomic conjugate, is the inverse of P. It is commutative and inherits the group properties closure and associativity from ordinary multiplication.

The triangle plane is divided into four regions by the four sign possibilities inherent in three homogeneous coordinates, which we notate as “o” (for original), “a,” “b,” and “c”. If P and Q are both in the same region of the triangle plane, P•Q is in the central region. If only one is in the central region then P•Q is in the same region as the other. If P and Q in different ones of the A, B, C regions, then P•Q is in the third. This group (not the Kimberling group) is shown in Fig. 3. For strictly affine considerations, this group lets us “mod out” coordinates with negatives and focus almost entirely on the central region. For non-affine structures this would be improper.
Most points are composite under this operation. For example the circumcenter O ~ :b² S_B: is the product of K ~ :b²: and tH ~ :S_B:. Some points, such as the incenter Io ~ :a: and the orthocenter H ~ :1/S_B: cannot be composite, except in trivial senses. They become the fundamental points out of which the others are created. It is not useful to call them prime because they would not be fundamental using other ways of defining the group operation (trilinear coordinates, for example).

The set of points {Pⁿ} where Pⁿ = P•P^n-1 with n an integer is the orbit of P. Figure 4 shows the orbit of Io, which includes many well known points. The “ellipse” formed is striking; there is too much structure to ignore1.

Even powers of P are the same for the four points (±x: ±y: ±z), so that the harmonic associates of a point share the same central orbit for even powers of P and are external for odd powers [Figs. 8, 9]. This means that each central orbit is related to three others consisting of the harmonic associates of points on the orbit. The central orbit of Io and one of the external ones is shown in Figure 4.

Properties of Orbits

Figure 8 shows the orbits for an arbitrary central point P ~ (x:y:z). The orbit is the set of points {Pⁿ ~ (xⁿ:yⁿ:zⁿ)} . The points of the orbit will turn out to be on well defined curves, called the path of the orbit.

(1) The orbit of the centroid G is itself. The orbit of an harmonic associate of the centroid consists of itself and G.

(2) The orbit of a point on a median is on that median. If on an exmedian, the orbit is on the exmedian and the corresponding median.

(3) For any P not on an edge, the orbit goes through the centroid, since ( x⁰ : y⁰ : z⁰ ) is in the orbit.

(4) In general points in an orbit accumulate at two vertices. If the three coordinates of P are unequal, then higher positive powers make the large relatively larger. Eventually one coordinate dominates and the points collect at the corresponding vertex, moving in along the edge corresponding to the smallest coordinate. The negative powers of the Pⁿ make a different coordinate the largest (the one that was smallest before) so that the side of negative powers (the isotomic conjugates) approaches a different vertex, along the side that corresponds to the first vertex. Hence in general the curve on which the Pn lie goes through two vertices. Figure 5 divides the triangle plane divided into regions separated by the medians. The relation between the normalized coordinates is shown for each region. The path of the orbit will move through regions of the same color. If P is external to the triangle, its central hamonic conjugate determines the vertices at which the orbit accumulates.

(5) In general the orbit is tangent to two sides of ABC. This follows from the last item and is a property of the curves, introduced below, that are the orbit paths. The exception is when the generating point is on an edge or a median.

(6) The exterior segments are the harmonic conjugates of the interior ones. If a point begins with any negative coordinates, even powers will be positive and be on the orbit of the corresponding central point [Fig. 6], odd powers will be harmonic associates of points on the central orbit.

(7) Orbits are isotomically self-conjugate in that if P is in the orbit, so is tP, its isotomic conjugate.

(8) Orbits are affine invariant in that any affine transformation takes an orbit to its equivalent orbit. Of course the interpretation of the orbit, as in “the orbit of the incenter”, is not usually invariant.

The group operation has a nice regularity property. Consider the Cevian rays from a vertex to points G, P, P² = P•P, P³ = P•P², as well as P^-1 = tP [Fig. 7]. If P ~ (p:q:r) then position of the A-trace of P on BC is determined by the ratio q/r. The traces of Pⁿ are determined by (q/r)ⁿ, which, if q≠r, are regularly spaced on BC with an accumulation point at a vertex.

These lines can be indexed by the exponent n, which is positive on one side of a median, negative on the other. The median has index zero; the P-Cevian has index 1, and the isotomic P-Cevian has index -1.

If Q is on one of these lines, then P•Q is on the next line (index one higher) in the same direction as G to P. Hence if Q is on the P-isotomic line, then P•Q is on median. These statements follow from the coordinates as well as from the indices.

If we repeat this from all three vertices we get an affine invariant grid which gives a nice way to construct, exactly or approximately, the position of P•Q (shown in Fig. 8) from the position of Q. This grid can be used as a coordinate system dependent on P. In this system G would have coordinate (0,0,0) and P would have coordinates (1,1,1) since it’s index from each vertex is +1. The index coordinates of Pⁿ are (n,n,n). Points not powers of P can also be given coordinates (not necessarily integers). The nice thing about this system is that multiplication by P will always add one to each index coordinate; multiplication by P-1 will subtract 1, and so on. This will be useful later.

The exact index coordinates of Q(p,q,r) with respect to the P(l,m,n) grid are ( log p/log l, log q/log m, log r/log n. Note: log(pqr)/log(lmn) is a distance function for the grid coordinates of points in the central region, a fact which is noted here but not used in this paper.

In most cases we will only use the grid lines from two vertices, and hence two coordinates, which suffice.

We are fortunate in having some ellipses which contain an orbit; i.e., if P is on the ellipse then so are all P_ⁿ. They are the very interesting y_² – zx = 0 and its cousins [Fig. 9]. This ellipse is a translation of the Steiner ellipse and centered at (2:-1:2), which is on the Steiner ellipse, and is tangent to the a and c sides of the triangle [3,4]. It is invariant under

(1) any affine transformation (since its barycentric equation does not include any triangle distances or angles)

(2) (x:y:z) implies (x:-y:z), the B-harmonic associate.

(3) (x:y:z) implies (1/x:1/y:1/z) self-isotomic

(4) (x:y:z) implies (x_ⁿ:y_ⁿ:z_ⁿ); For positive integral n, y_²ⁿ – x_ⁿz_ⁿ = (y_² – zx)( polynomial of degree 2n –2) so that if P is on the ellipse, so is P_ⁿ. This is still true if n is negative since then 1/y_² – 1/zx will be a factor.

This last property establishes that all points on the orbit of P lie on this ellipse if P does.
If a coordinate of P is negative, then even powers are positive so the orbit of a point outside the triangle bounces back and forth between the central orbit and its harmonic associates. There are three ways this can happen, one for each coordinate, giving three outside harmonic conjugate curves. One of them completes the ellipse, and the other two form an hyperbola with asymptotes.

The harmonic associate of a point on the medial triangle is at infinity, hence there are two asymptotes, one for each side of the medial triangle that the ellipse crosses [Fig. 9].

The A and C harmonic associates of y2 – zx = 0 are the two branches of the hyperbola y2 + zx = 0, centered at ( 2 : 1 : 2 ), itself an harmonic associate of the ellipse center (2 : -1 : 2 ).

The intersections of the c side of the medial triangle (z = x + y) with the ellipse are the two points ~ ( 1/ g: 1 : ) and Bt ~ (1/g : -1 : ) where g = (1 + √5)/2 is the golden section making both C and CBt parallel to an asymptote.
The intersection of the a side of the medial triangle (x = y + z) with the ellipse is the two points t ~ ( : 1 : 1/ ) and Bt ~ ( : -1 : 1/) and which are isotomic to the above two. At and AB are each parallel to an asymptote.

The asymptotes are parallel to these lines through the hyperbola center ( 2 : 1 : 2 ).

Theorem: The points ( xⁱ , yⁱ , zⁱ ), with x,y,z positive, normalized and i an integer, all lie a median or a curve of the form y_ⁿ = zx.

If two of x, y, and z are equal then those coordinates will be equal for all powers, placing all orbit points on a median. If P = ( p , q , r ), with p, q, r unequal and normalized, is on the curve and ( x , y , z ) = ( pⁿ , qⁿ , rⁿ ) is a general point on the curve then

Using base q for the logarithms where q is middle in size compared to p and r, we have

or
.
telling us that y_ⁿ = zx with n = logq/pr. Of course the curve might be xn = yz or zn = xy; the position of P determining the curve.

This curve is the property of the affine point with coordinates ( p : q : r ) and is affine invariant. The family of all possible orbits can be thought of as affine invariant “field lines” [Fig. 10] controlling the relative positions of points. yn + zx = 0 is a companion graph containing the A and C harmonic associates of points on the internal orbit. This graph will have asymptotes determined by the intersections of y_ⁿ – zx = 0 with the medial triangle.

These curves have properties similar to y_² = zx. They are tangent to the sides at two vertices.
Surprisingly the intersection of these non-homogeneous nth degree curves with themselves or any straight line can be solved (by the Mathematica oracle), but the solutions are not points we normally see in triangle geometry.

Central orbits organize a group of points together on a known curve. Those of P and Q give us two curves of points originating and ending at vertices, each going through the centroid and each with its own affine grid. But an important feature of the orbits of a group is that composite points also arise from them. Hence from the orbit of Io ~ :b: and tH ~ :SB: we can find b SB, b_² SB, b_²/SB, and similar very well known points. A group can be useful, not only by organizing points, but letting us do mathematics with those organizations.

The obvious way to proceed is to examine what happens when a point Q multiplies every point of the orbit of P {Figure 12], resulting in a sequence of points through two vertices. Q·orbit(P) = {Q•P_ⁿ} , goes through Q instead of G. Similarly this orbit’s path is tangent to two edges and points accumulate at the same two vertices as orbit(P).

If the orbit of P is y_ⁿ = zx and if Q = (l,m,n), then Q•orbit(P) lies on (y/m)n = zx/nl, for which it can be seen by substitution that Q•P_ⁿ satisfies this equation.

A key feature is that the regularity of the grid is maintained [Fig. 12], which can be seen by noting that the multiplied points have the same relative position to the rays through points of the orbit of P.

Index coordinates work nicely here. If any Q is multiplied by P, then the index coordinates of Q, in relation of the P-grid, are each increased by 1. If by P_ⁿ the index coordinates increase by n.

An orbit times an orbit

Now the next step: we multiply orbits. Each member of an orbit multiples each member of the other orbit [Fig. 13]. We give this the notation P X Q = {P_ⁿ•Q_^m} with n,m Œ , which is shown in Figure 12 where the grid through the Q orbit is added for reference. Note that the grid, determined from one orbit, helps position the whole lot. If many known points are represented in such a grid, their relative positions can be understood, at least approximately. Figures 9, 10, 11 give the composite structures for Io X H, Io X No, and Io X S where No is the central instance of the Nagel point and S is the Steiner point. In each case we recover many known points. The regularity of the spacing is what gives this structure importance.

In Figure 13 the grid is determined by the points of the Q orbit, but the other points conform to it as well. In the figure we may estimate that the index coordinates of P are (2.2, .3). The point P2 has double these coordinates, as can be seen from the figure. Each time a point is multiplied by Q, we add (1,1) to its grid coordinates, making it maintain the same position relative to the grid lines. Thus P•Q has grid coordinates (3.3, 1.3) as can be seen in the figure.

Often it is useful to consider that P•Q has sub-orbits: G•orbit(Q), P•orbit(Q), P2•orbit(Q), etc. These can often be indexed from the point not used to draw the grid. This is particularly useful when one or more of the points is exterior to ABC.

yn ± zx and the non-Euclidean distribution of Euclidean triangle centers

yⁿ ± zx and the non-Euclidean
distribution of Euclidean triangle centers