Points on G—P,
and related conics

(revised 9-01-06)

The goal is to understand points on an arbitrary line G—P through the centroid. There are an infinite number of points on this line, most are not symmetrically defined centers. These two conditions: centeredness and being on this line, have the right amount of freedom and restriction to give interesting results.

Properties of G—P:
the directions of the two lines G—P and ~P, the dual of P (see here for this notation), map to antipodal points on the Steiner ellipse: t(∞•~P), t(∞•(G—P)) The dual of P is written ~P.

The dual of each point on G—P is parallel to ~P.

The medial and dilated operations. Centrality.
P is assumed to be specified as a construction applied to triangle ABC. If this construction leads to the same point when A, B, C are cyclically permuted, we say that P is a central point. The centroid is central but the midpoint of any edge is not. mP is the result when the construction that produces P is applied to the medial triangle, dP when it is applied to the dilated (antimedial) triangle. Since these two triangle are dilations from G of ABC, mP and dP are on the line G—P. The points P, G, mP, dP give rigid structure to the line, the same for any triangle.
Note: the m and d operations may be repeated, always resulting in a point on G—P.

A special point m(P2-) = (: n2 – lm + l2 – mn : ) is the intersection of the dual of P with G—P. Its conjugate is the fourth intersection of the P circumconic with the Steiner ellipse. We call it the weakened Tarry point. Note: b2– = b2–ca; P2– is the Steiner inverse of P. Note: for central points, we only give the middle, or central, coordinate.

This topic is shown in 5 pictures. The first shows the lines G—P, ~P, and the points on the Steiner ellipse that correspond the directions of the two lines. Coordinates are in red. The Steiner ellipse is shown in each picture.

In these pictures P is Io, the incenter, which is perfectly general since the incenter carries the affine information of the triangle. Notation is here. The first picture shows the most important points on G—P. mIo = So, the original Spieker point, and dIo = No, the original Nagel point.

The "weakened" operation w

If, in the barycentric coordinates of a point, we replace a2, b2, c2 with a, b, c, we get a different version of the point that has many of the same constructive properties. We call this operation "weakening" the point, denoted by the prefix w. Its inverse s, does the opposite.

Note: s is usually 4 to 1, but w is 1 to 4.

Noteworthy points
The infinite point ( : c+a – 2b : )
The Spieker and Nagel points, the medial and dilation of the incenter, respectively.
The proSpieker point ( : b2(c+a) : ) which is the center of the following homothety:
The Steiner inverse of Io and its medial, which is the Ur-version of the Tarry point structure.
Intersections with the Steiner ellipse, listed as Q1 and Q2 in picture 3, which lead to the asymptote directions for the hyperbola below.

Harmonic conjugate pairs (notation on chart below).

Io, So; G, No
G, Io ; P4, P5
G, Io ; P5, P6

Points as Prime

The coordinates of a points of a point on G—P may be used as inputs into other points. For example if ( : sb : ), which is on G—P is substituted into (: b2(c+a) :), also on the line, we get ( : b sb2 :), yet another point in G—P. The point coordinates, than thus be combined under composition to obtain more points on G—P.


Figure: The dark blue lines are G—Io and ~Io. The Steiner ellipse is shown with the isotomic conjugates of the endpoints of the two lines, which are antipodal. Centers from the following table are shown. These points are like primes in that they generate most of the other centers.

Points on G—Io
name middle coordinate comments
P0
:1:
 centroid
P1
Io = :b:
 incenter
P2
∞•(G—Io) =
: c+a–2b :
 infinite point
P3
~So•(G—Io) =
: b2–ca :
 Steiner inverse of Io
P4
: b2(c+a) :
 P0,P1; P4, P5 harmonic conjugates;
the pro-Spieker point;
a center of homothety*
P5
:b(ab + bc – ca):
 P0,P1; P4, P5 harmonic conjugates;
P6
b(ab + bc – 2ca)
 P0,P1; P5, P6 harmonic conjugates;
where tripolar of Io meets G—Io
b sb2
 = P4 of (No=dIo)
P7
(c+a)SB
(c+a–2b)3
 = P4 of P2
= P7 of P2
(a+c)2(a+2b+c) = P4 of mP1
(a+c)(a2–2b2+c2) =P6 of mP1
(c+a)(SB±const) harmonically conjugate with P7 and So = mIo;
related to Fermat points if constant = Sπ/3

Chart: gives middle coordinates of points on G—Io. The left column identifies ones that can be considered prime in that the others can be derived by composition. The right column gives comments, including how they are composed of the "primes," which can be seen to not be unique.

Notation: I use Conway notation where sa = s – a, sab = sa sb, and SA = (b2+c2 – a2)/2, and SBC = SB SC.

The formulas above should be thought of recursively, that each is a pattern which can be applied to all the others, including itself. For example the medial operation m, one of the operations that takes points on GP to points on GP, can be applied many times as in mmmP, which will also be on GP.

I checked my work by looking for the points on G—Io in ETC. I am trying to find an acceptable starting set of points to generate the rest in the way that primes generate the integers. Note that doing this for the incenter Io = (a:b:c) is equivalent for doing this for a general P = (l:m:n). The point Io implicitly contains all the affine structure of the plane (the reason this point is so interesting!). Simply replace (a, b, c) with (l, m, n) to find the points for the general G—P line.

The next picture adds the conics that are the isotomic conjugates of the two lines. If P were K, the symmedian point, these two conics would be the circumcircle and the Kiepert hyperbola. In our case it we have the weakened versions of both, the Io circumconic, and the weakened Kiepert hyperbola.

This conic has perspector ~P•∞ at infinity. Conics with perspectors at infinity have special properties.

 

The next picture shows relevant lines, such as the asymptotes of the hyperbola, related to the intersection of G—Io with the two ellipses.

 

Add inconics

The dual of a circumconic, perspector P, is an inconic, perspector tP. In particular the dual of a circumconic with perspector at infinity is an inconic with perspector on the Steiner ellipse, which is known to be a parabola.

The Kiepert Parallelogram

John Conway noticed that the points GHUT form a parallelogram, where U is the intersection of the Kiepert hyperbola with the Steiner ellipse, and of course, T, the Tarry point, is its intersection with the circumcircle. The affine theory tells us that this parallelogram is universal, as shown below for the weakened Kiepert hyperbola (formed from Io rather than K).

Also added to this picture are lines that show that the weakened Steiner and Tarry points behave like their more famous namesakes. wS and wT are opposite in the Io-circumconic center (this is the weakened circumcircle). wM = w(mS) is the weakened Kiepert center.