The equivalence class of the Fermat points

More from the mining the Kimberling points project. I am trying to separate his points into equivalence classes (or perhaps cosets). This note is about points whose coordinates contain the square root of three. I am also trying to show how effective this separation is.

At the end of this page is a list of all points that have a root3 in their coordinates.

These are points that John Conway calls "semi-strong" because they are always on a strong line. The fact that they never occur on a weak line is not required by nature, yet never realized in nature.

The Fermat points, the isodynamic points, the Napoleon points all have a factor of  S_pi/3 =  2 root3.area in their coordinates. This means that (1) they are separated algebraically from the great mass of geometrical points, which do not contain this root and (2) that there is a group of points with this root that they will relate to.

As an example here are the lines listed in ETC on which the Fermat point resides. I have colored each point that has the root3 factor in green. Strong points are blue and weak ones are red.

The first thing to notice is that all  these combinations have at least one green point. This is a list of colinearities. Algebraically this means that the coordinates of X13 are linear combinations of the stated point. Since x13 has root3 in its coordinates, at least one of the other point has it as well. In the paragraphs on the isodynamics, I will talk about the use of of strong and weak points.

X(13) lies on these lines:
2,16   3,17   4,61   5,18   6,14   11,202   15,30   76,299   80,1251   98,1080   99,303   148,617   203,1478   226,1081   262,383   275,472   298,532   484,1277   531,671   533,621   634,635

Here is the same information for an isodynamic point

X(15) lies on these lines:
1,1251   2,14 3,6   4,17   13,30   18,140   35,1250   36,202   55,203   298,533   303,316   395,549   397,550   532,616   628,636

This listing of points is more interesting because of two particular listings.

x15 is on the strong line 3,6 . There should be one such listing for each point of this nature. The equivalent for x14 is 2,14 which has other strong points not listed.

The two isodynaics are of the form  in,s =  : bb SB ± bb Spi/3 : . These four points are harmonic conjugate pairs

           in,     in+is ~ O,    is,      in - is ~ K

This method gives the strong line on which the isodynames reside, which is OK. This explains the only listing with no green points, ie, ones with a root3. Others are possible, but not listed.

The listing that shows that     13, 1, 1251 are colinear is very strange.  X1251 is a point you never heard of named after a star you have never heard of, the ultimate in whimsical (and meaningless) point naming. But algebraically it is interesting. Its coodinates contains root3, but also depends on sin A/2, which is a messy square root in the triangle sides ( square root of a square root). The fact that it would be colinear with the incenter and the isodynamic point is so strange that I plan to investigate this pairing.

On yes the fn--fs line, the Fermat join, contains the strong points of the midpoint of GH and the center of the Kiepert hyperbola.

The list. All these points have a √3 in their coordinates. They are an equivalence class in the Kimberling Xpoints.

13
14
15
16
17
18
61
62
202
203
299
300
301
302
303
383
395
396
397
398
463
465
466
470
471
472
473
530
531
532
533
554
559
616
617
618
619
633
634
635
636
1080
1081
1082
1094
1095
1250
1251
1276
1277
1493
1510
1511
1652
1653
2962
2963
2964
2965

Created by Mathematica  (December 11, 2005)