The Triangle is a Busy Place —
The Distribution of Triangle Centers Project
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Clark Kimberling's Encyclopedia of Triangle Centers lists over 3000 triangle centers. The existence of such a large set of points together with an easy way to get their coordinates into a computer allows us to see and think about triangle centers in new ways. Many have contributed to this project including
Edward Brisse, who translated ETC points into barycentric coordinates. Here are all 2400 points from one of Edward Brisse's compilations of points into barycentric coordinates. Edward took Clark Kimberling's Encyclopedia of Triangle Centers and converted them to Barycentric coordinates (here is that data set.
Nikolaos Dergiades, who compiled the points into the spreadsheet feature of an older version of Geometer's sketchpad. These pictures were not dynamic but the pattern of the points could easily be seen.
Peter Moses, who developed a way to plot a point in Sketchpad using homogeneous coordinates. Peter has also compiled much information about ETC points on lines, circles and conics.
Jason Cantarella of the Univerity of Georgia mathematics department, who put the coordinates of ETC into Perl, available here. You do not have to use Perl to find this useful, just use a text editor to clean the file to the format you prefer.
Nick Jacqiw, the inventor and devoloper of Sketchpad became interested in this project. Yao Liu, working as a sketchpad developer, has turned Peter's work into tools to plot points by their barycentric coordinates. His goal was to make a dynamical version of with all 3000 points. Currently he is up to 360, which Yao says takes us up to the second edition of Kimberling's book.
What we can learn from this project
Dynamical behavior
We tend to look at properties of single point. What line or circle are they on; how are they constructed; is it a perspector. But each point has dynamical behavior in its motion as the reference triangle is changed. The motions of different points have common behavior. Hence dynamical behavior can be a topic for mathematical discussion.
The large scale structure of many points.
The most obvious of these is the sweeps phenomenon, discussed here.
The relation of particular families of points to the whole.
The triangle points can be divided into equivalences in many ways. The most famous is what John Conway calls extraversion, but one can think of others as well: points on the circumcircle, points with SB (ie cos B) as a factor in the b-coordinate, and so on.
Rigid versus non-rigid structure.
This is very important. We measure structure of new objects against known structure of old ones. The grear examples of this are the Euler line, where we know that O, G, N, H will be on this line with a given spacing. The circumcenter of the tangential triangle is also on this line, but who knows where as it moves wildly up and down the line (see this movie of point on Euler line). The discover of geometrical stucture that maintains its coherence in all circumstances is of vital importance. Often rigid structure can be indicated as here.
To facilitate this discussion I have had to invent a number of new terms. The web links take you to illustrations of the phenomena.
Sweep: Triangle centers are not uniformly distributed in the triangle plane. They cluster around a favored path through the triangle interior. I call the distribution of these points "a sweep," related to the invariant path of the projective transformation represented by barycentric multiplication by Io, the original incenter. Here is a web page on sweeps.
The affine parallelogram of H: The lines HK, DO, HdK, DL form a parallelogram. (Affine invariant form: P mtP, tP mP, P dtP, tP dP). Here t, m, d are the isotomic conjugate, medial and antimedial (dilated) operations). D = tH = dK and L is the deLongchamps point. This particular parallelogram seems to be related to the positions of aggregates of triangle centers.
Corresponding points on two conics determine a line that goes through the fourth intersection of the conics.
Weaknesses of this project
The weaknesses are those of the ETC itself. For many points (I estimate about 1000 of the 3000) their only relavance is that they are in the list, generated by obscure or irrelevant transformations. See here for a discussion of this.
Here are the first 260 ETC points plotted by Yao Liu using their barycentric coordinates. I have colored the strong points blue and weak points red. The Kipert hyperbola is red, the Jerebek hyperbola is blue. The circumcircle, 9-circle, and (H, 2R) circle are shown. The Steiner ellipses are shown. The Euler, Brocard, and triangle edge lines are shown.
Figure: this shows the shape of the triangle that puts the maximum number of points in the sweep area. Note that a large number of points are inside the affine parallelogram of H which is light yellow. The Kiepert hyperbola is red, the Jerabek hyperbola is blue. The number next to each point is its index in ETC.
Analysis of this picture is here. Having dynamical documents with 260 points is invaluable. It gives one a whole new view of our subject. Here is the Sketchpad file (of 260 points).
Here is a movie from which the above picture is taken. It is in iPod size and format and costs 7 MB. The movie is created by rotating the A vertex around a circle. At times the triangle will be right, isosceles, acute, obtuse and almost equilateral. This movie shows the first 360 ETC triangle centers.
What is interesting here is the flow of the points. Most follow the pattern that I call the sweep of the incenter but many points flow into and out of this pattern. The two hyperbolas seem to shepherd the points and form boundaries for some of them. Most of the points are inside the affine parallelogram of H (yellow). The Jerabek hyperbola degenates at right and isosceles triangles; Kiepert degerates at isosceles. Towards the end of the movie the triangle becomes almost equilateral (see below); look for the three points that follow a deltoid. This movie is neat! There is so much more activity than one would expect.
A more detailed analysis is here. The pattern that I call "the sweep of the incenter" is clearly shown (see illustration below). I created the Mathematica pictures below to check this prediction, which comes from extra-synthetic methods. See also here for explanations of this. More about this below.
First is the pattern for an acute triangle; then the pattern for an obtuse one. Click on the pictures for a much larger view.
Degeneracies and special triangle shapes
Dynamical programs allow us the very useful diagostic device of studying degeneracies. For example, if a point coordinate contains the SB factor (ie, cos B), its behavior for right triangles, when this factor is zero, can be diagnostic. If the b factor (sin B), highly obtuse triangles will be diagostic. Yao Liu found many interesting things about almost equilateral triangles.
Yao Liu's observations: nearly equilateral triangles
Yao's comments:
Sketchpad shows that the first 282 centers (minus X245 , X246 , X247 ) of ETC can be grouped
in 9 groups. In order to describe where they are located more precisely, we can take note
of a few structures. Circumcircle and nine-point circle are well-defined and well-behaving
in the neighborhood of e. Euler line and Brocard meridian tend to coincide, and their
isogonal conjugates, Jerabek and Kiepert hyperbolae tend to coincide. I’ll call them the
line, and the hyperbola, avoiding favoring one name to another.
I. 191 of them are in the neighborhood of the center of the equilateral triangle.
II. 15 are at the (fourth) intersection of the hyperbola with circumcircle. [the Tarry point]
III. 17. medial of group II. On nine-point circle.
IV. 5. dilated of group II.
V. 11. Antipode of group II in circumcircle.
VI. 15. medial of group V. On nine-point circle.
VII. 3. dilated of group V.
VIII. 3. Mysteriously on a deltoid with vertices ABC: X59 , X249 , X250 .
IX. 19. Far-out on the line.
Figure: Point distribution for the almost equilateral triangle. The A vertex is very close to the position that makes the triangle equilateral. Most point collect in only a few spots enumerated in the table by Yao Liu above. Here X59 lies on an deltoid whose vertices are A, B, C.
The Kiepert, Jerabek, and Feuerbach hyperbolas
What I have been looking at is the set of points on the Kiepert (KH), Jerabek (JH), and Feuerbach (FH) rectangular hyperbolas. All three are related to both the affine parallelogram and the sweeps phenomenon. The path of the Kiepert and Feuerbach hyperbola is along the path of the sweep. The Jerabek hyperbola is attatched to to the the edges of the affine parallelogram, which helps define the sweep area. Pictures of this can be seen on my web pages.
Rigid behavior on these hyperbolas. This chart shows a group of points that keep a rigid relationship to each other. All are inside the H affine parallelogram, two of whose vertices are 4 (H) and 69 (D = tH). Vertical points are corresponding, in that the line between them goes through H. Horizontally the points are always on their respective conic in the order listed.
KH 4 226 2 10 321 76
FH 4 1 21 9 8 314
JH 6 73 3 ? 72 69
This connection of this rigid behavior with the affine parallelogram is dramatic as can be seen in the picture shown here.
Figure: Corresponding points on the Kiepert (blue), Jerabek (red), and Feuerbach (green) rectangular hyperbolas which are connected by lines going through H, the fourth intersection of each pair of conics. The H-affine parallelogram is shown in blue. The path traversed by the Kiepert and Feuerbach hyperbolas is the "sweep." These points are the ones listed in the table of points below.
Synchronized behavior of points on the hyperbolas.
A nice thing to do with dynamical pictures is to look at the motion behavior of points on well known objects. Here we look at points on the Kiepert and Jerabek hyperbolas. If you look at both at once (see movie below) it becomes obvious that the behavior of many points on the two hyperbolas are coordinated, meaning that as the triangle changes shape the movements of each is highly correlated with the movement of its pair. The picture shows three pairs of coordinated points.
On the left of the picture are two coordinated pairs of points, one strong (blue) and one weak (red). As the shape of the triangle changes these point move along the hyperbola in a coordinated way. If one meets a vertex, the other will as well.
Likewise X94 and X265 will meet the vertex at the same time and be coordinated in all their movements.
The line joining coordinated points, one on each hyperbola, goes through H, the fourth intersection of the two hyperbolas. This is explained below.The O-projective transformation
The y coordinate of the perspector of the Jerebek hyperbola is b2SB times that of the Kiepert one. Hence any point on Jerebek can be obtained from a corresponding Kiepert one by barycentric multiplication by (:b2SB:), the coordinates of O, the circumcenter. Barycentric multiplication is equivalent to a projective transformation that preserves ABC. For conics this transformation has a nice implementation: a line from the fourth intersection of the two conic connects corresponding points on the two conics.
For the Kiepert and Jerabek hyperbolas the fourth intersection is H. The lines from H through corresponding pairs of points are shown in the figure.
The following movie shows the motion of points on the three hyperbolas. It is in ipod format and is 2 MB in size.
Figure: Points on the Kiepert (bllue), Jerabek 9red) and Feuerbach (grreen) hyperbolas are shown here. Corresponding points on the three conics are connected by a dotted line, which goes through H.
Points on the circumcircle and 9 point circle
Here is a move showing the behavior of these point. The important thing to notice is the coordinate motion of many of the points. This kind of rigidity can be used to measure the behavior of non-rigid points.
Correponding points on KH and JH and FH
A point at infinity can be mapped to corresponding points on these three famous rectangular hyperbolas. The line joining any pair of corresponding points goes through the fourth intersection of the two conics. Since these conics all go through H, this is their fourth intersection.
This chart shows corresponding points on the three hyperbolas. In each case the line joining them goes through H. Corresponding points have correspoinding dynamical behavior. For example if one goes throught a vertex, they all do. If one has rigid behaviour, they all do.
The colored regions in the chart classify the infinite points by their origin.
∞•~Io
So
Mo
twS = ∞•~tIo
No
∞•(Io—tIo)
∞•~Go
Io
∞•~Go2
Go
?
tS = ∞•~K
G
O
Shiffler Ho
gS = ∞•~tK
R = tK
D = tH = dK
rHo
~H infinity
H
K
?
rT
Tarry point
?
?
?
Tarry point
?
?
also the Circumcenter
∞•~H = ∞•~O
H
K
∞•(G—H)
infinite point on Euler line
Fermat
b2SB/(SB+Sπ/3)Fermat-like point on Jerabek
:1/(SWB + constant SCA):
Figure: This shows most of the 3000+ points in ETC. Some point are outside the viewing area.
Here are some other pictures by Nikolaos, 500 points at a time. Gives good insight into how ETC is structured.
These were done by me using Edward Brisse's compilation of over 2000 ETC points, which did not include points at infinity or points whose coordinates have square roots such as the Fermat points.
Here is the pattern for an obtuse triangle. The sweep of the incenter is even more pronounced.
Pictures pasted into a database program Fathom
This uses data supplied by Nikolaos Dergiades. There are two "parabolas" these pictures whose existence is puzzling.
The second picture has nice detail of the triangle interior.
The sweep of the incenter
This pattern is shown as a dark blue band through the triangle interior. To understand this pattern try these pictures and the links therein. Here is a Geometer's Sketchpad file showing the orbit of a point.
A sweep related to infinite points of lines derived from the incenter can be clearly seen as well.