CoNormalization, an equivalence relation among points
Now that we know thousands of triangle centers and thousands more non-centers, we can try to find patterns in those points. One way is to find equivalences where points can be divided into equivalence classes. Extraversion is one of these, but there are more. Here I show a way of dividing triangle centers into equivalence classes based on the sum of their coordinates, but rather than work on particular points, we proceed in the most general way.
We generate points in an invariant way, beginning with the 3 points: the centroid G and an arbitrary point P and its isotomic conjugate P' = tP. The three operations t, the isotomic conjugate; m, the medial operation; and d, the medial inverse are used succesively on each point generating the structure of points in the first figure. This is clearly a symmetric stucture. The ancients called this the anallagmatic symmetry by which a point can be exchanged with its conjugate. If we choose P = incenter, we reproduce much of the structure of ETC, but P = Nagel or P = orthocenter are more interesting choices.
To these points we may add the intersections between the lines connecting them. The resulting structure of points will be both affinely and anallagmatically invariant. This page is a restricted version of a program explained here.
Normalization
When we use the homogeneous form of points (using the colon as the separator between coordinates), we determine projective properties such as concurrance and colineation. To determine the relative position of these points, an affine property, we need to use the normalized coordinates, which add to 1. We will indicate the sum of the coordinates of P = (l:m:n) by T = l+m+n. Because T can be considered a fourth coordinate, we will speak of “the total coordinate.”
Let C1, C2, C3 be the homogeneous coordinates of three points, with C1 = C2 + C3, making the points thus represented colinear. P1 = C1/T1 is one of the points written with normalized coordinates. The operation for the total coordinate is linear, so their total coordinates have the relation T1 = T2 + T3. The normalized points can thus be written (T1 + T2)P3 = T1 P1 + T2 P2, or P3 = T1/(T1+T2) P1+T2/(T1+T2) P2. The relative positions of the points is then shown in this diagram.
If points have the same total coordinate, we call them co-normalized and the algebra works out particularly well. It is equally as nice if the total coordinate of one is an integral mulltiple of another. All such points form an equivalence class under conormalization.
For a center, defined as a point whose coordinates are related by cyclic permutation of parameters, the total coordinate is a symmetric function of those parameters. Hence it is the symmetric functions of parameters that determine the placement of triangle centers on lines.
Since the sum of the three coordinates of mP is twice that of P, the point P is twice as far from G as is mP.
A more substantial example is the positioning of tmP on the P—P' line. Using middle coordinates only, tmP ~ :(l+m)(m+n): = :m(l+m+n) + nl:, which shows that tmP is colinear with tP and P and that the spacing of tmP on this line is controlled by the ratio (l+m+n)2/(mn+nl+lm).
Conormalized third degree points
Here is a set of points in the same equivalence class under conormaliztion. We create third degree expressions for some points by adding symmetric factors of the appropriate degree, which will be the same for each coordinate. For each point we give the b-coordinate and the total coordinate, showing the nice result that the total coordinates are all the same up to a factor.
|
point
|
b coordinate
|
total coordinate
|
|
G
|
(l+m+n)(mn + nl + lm)
|
3(l+m+n)(mn + nl + lm)
|
|
P
|
m(mn + nl + lm)
|
(l+m+n)(mn + nl + lm)
|
|
tP
|
nl(l+m+n)
|
(l+m+n)(mn + nl + lm)
|
|
mP
|
(n+l)(mn + nl + lm)
|
2(l+m+n)(mn + nl + lm)
|
|
mtP
|
m(n + l)(l+m+n)
|
2(l+m+n)(mn + nl + lm)
|
|
dP
|
(l–m+n)(mn + nl + lm)
|
(l+m+n)(mn + nl + lm)
|
|
dtP
|
(mn – nl + lm)(l+m+n)
|
(l+m+n)(mn + nl + lm)
|
|
mmP
|
(l+2m+n)(mn + nl + lm)
|
4(l+m+n)(mn + nl + lm)
|
|
mmtP
|
(l+m+n)(mn + 2nl + lm)
|
4(l+m+n)(mn + nl + lm)
|
|
ddP
|
(l–3m+n)(mn + nl + lm)
|
–(l+m+n)(mn + nl + lm)
|
|
ddtP
|
(l+m+n)(mn – 3nl + lm)
|
–(l+m+n)(mn + nl + lm)
|
|
Q
|
m(l2 + nl + n2)
|
(l+m+n)(mn + nl + lm)
|
|
dQ
|
–l2 m + l m2 + l2n + l m n + m2n + ln2 – m n2
|
(l+m+n)(mn + nl + lm)
|
|
15
|
2 l m2 – l2 n + l m n + 2 m2 n – l n2
|
(l+m+n)(mn + nl + lm)
|
|
21
|
– l m2 + 2l2 n + l m n – m2 n + 2l n2
|
(l+m+n)(mn + nl + lm)
|
|
29
|
(l + n)(l m – m2 + 2 l n + m n)
|
2(l+m+n)(mn + nl + lm)
|
|
30
|
(l+n)(l m + 2m2 – l n + m n)
|
2(l+m+n)(mn + nl + lm)
|
The numbered points refer to this chart. Column 2 is a new way to notate a point. I call it "constructive notation", because it indicates one way to create the point, given knowledge of P in respect to ABC.
There are new points that arise as intersections, and are not related to P by a series of affine oparations. These points are so interesting that they warrant a section of their own.
The new points
10 of the 17 the points in the above table are defined as intersections. Of these, 4 are at infinity, epressing native parallelisms, but 6 are new and are considered here. In the figure below some of these points are made large and show their names inside the bounding circle. We will call these points Q = P23, dQ = P22, which are vertices of the affine parallelogram, and P15, P21, P29, P30 where the numbers refer to their position in the above points list.
Q, dQ, P21, and P15 have the same total coordinate (l+m+n)(mn + nl + lm) as above, while P29 and P30 have double this value. To have so many points that are conormalized (up to a constant factor) is serendipitous. a situation where simple algebra and simple geometry will conincide. These 17 points are each related to the others. This is so nice to see that we will work it out here. In this section only, expressions such as P15 will stand exactly for the 3rd degree coordinates given in the table. By any means the mutual interrelationships are delightful. We simply add or subtract the algebraic expressions for the various points, sometimes adjusting for the value of the total coordinate, and note the conclusions. We will show the relationships for the P15 point, the others are analogous and can be seen in the figure below.
Q–P15 = (2 l – m – n) (l m + l n + m n) ~ ∞•(G—P) hence, Q—P15 is parallel to G—P.
G + P15 = P29 hence, P29 = m P15 (its total coordinate is twice as big).
P15 + tP = 2 P hence, P is midpoint between P15 and tP.
P15 + 2 P21 = 3 tP so that P15, tP and P21 are distributed in the ratio 2:1.
dP + P15 = mtP so that mtP is the midpoint of dP and P15
dtP – ddP = P15 so that P15 is the midpoint of dtP and ddP, which has negative total coordinate.
ddP + ddtP = 2 dQ so that dQ is the midpoint of ddP and ddtP.
Figure: This shows the affine parallelogram of the point P (or equivalently P' = tP) and the interrelation of the 17 points listed above.
This procedure of conormalized equivalences can be extended to factors other than integers, potentially creating classifictions of all points. This is a big task and requires a computer; I have not implemented it yet.