The mathematician Clark Kimberling estimates that he spends between two and five hours each day, seven days a week, maintaining his online encyclopedia of triangle centers, which currently contains entries for 37,887 points related to the triangle. A strange way to spend one’s time, you may think. What is a triangle center anyway? And how could it be possible that from a triangle—just three points—come 37,887? Kimberling, a professor at the University of Evansville, introduces his encyclopedia as follows:

Long ago, someone drew a triangle and three segments across it. Each segment started at a vertex and stopped at the midpoint of the opposite side. The segments met in a point. The person was impressed and repeated the experiment on a different shape of triangle. Again the segments met in a point. The person drew yet a third triangle, very carefully, with the same result. He told his friends. To their surprise and delight, the coincidence worked for them, too. Word spread, and the magic of the three segments was regarded as the work of a higher power.

The mentioned point of intersection is called the “centroid” and is the second point listed in Kimberling’s encyclopedia. The centroid is the center of mass of a triangle. Cut a triangle out of cardstock, and it will balance on the tip of a pencil at its centroid. For this reason, the centroid is a very natural “center” of a triangle. But so are many others. Another often-considered triangle center is the circumcenter—the center of the circle that passes through a triangle’s three vertices. Another still is the Fermat point, for which the sum of its distances to the three vertices is minimal. In the 1990s, Kimberling formalized the definition of a triangle center by examining points as “functions” of the three vertices. With the definition established, a list of known centers could be compiled. First, Kimberling did so in a book, and when the list continued to expand, he created his online encyclopedia. In Kimberling’s notation, the first three triangle centers are X(1) the incenter, X(2) the centroid, and X(3) the circumcenter. In his honor, triangle centers are sometimes called “Kimberling centers.”

Around 300 B.C., Euclid established in his Elements five assumptions that form the basis of Euclidean geometry. The postulates appeared to be self-evident from the physical world (though Einstein would later point out this assertion’s limits). The second postulate, for instance, simply asserts that between any two points it is possible to draw a straight line—an uncontroversial statement. From the five postulates, one can prove the coincidence of the centroid. Because geometric theorems could be deduced from self-evident principles, early philosophers pointed to them as examples of certain truth. Plato said “The knowledge at which geometry aims is the knowledge of the eternal.” It is suggested that the centroid and other triangle centers exist intrinsically, independent of human construct. In this way, triangle centers, like stars in the sky, are discovered rather than invented.

The top of the website reads “Clark Kimberling’s Encyclopedia of Triangle Centers - ETC.” The last three letters, Kimberling says are an intentional pun. Below the title is a set of buttons for navigating the site, each fittingly in the shape of a triangle. The website is formatted in no-nonsense HTML. The information does the talking. In twenty parts—the nineteenth nearly filled, the twentieth ready to follow—known triangle centers are enumerated, each entry including a substantive list of properties. Scrolling down the encyclopedia, you are bombarded by blocks and blocks of numbers and text, the aggregation of knowledge apparent in their collective mass. The first part alone comes to 362 pages when printed in small type. Names of points often suggest their histories. X(20) is the De Longchamps point, discovered by the French mathematician Gaston Albert Gohierre de Longchamps. X(21) is the Schiffler point, discovered by Kurt Schiffler, a German engineer. X(22), the Exeter point, was discovered during a mathematics workshop at Phillips Exeter Academy in New Hampshire in 1986.

In 1798, Napoleon Bonaparte led a military expedition with the intention of seizing Egypt and impeding British trade routes. Napoleon, a newly elected member of the French Academy of Sciences, brought with him 167 researchers, including the accomplished geometer Gaspard Monge. Napoleon was an amateur mathematician and enjoyed keeping scientists in his company. Monge, who pioneered descriptive and differential geometry, has no center named after him, but Napoleon, it turns out, has two. X(17) is the first Napoleon point and X(18) the second.

The two Napoleon points arise from Napoleon’s theorem: Construct three equilateral triangles facing outwards upon the sides of a triangle; then the centers of the equilateral triangles form another equilateral triangle. The validity of the theorem’s name has been thoroughly questioned. Regardless of Napoleon’s mathematical inclinations, there is no evidence to suggest that he is connected to the theorem.

H.S.M. Coxeter and S.L. Greitzer write in their popular textbook that despite the recognition Napoleon receives for the theorem “the possibility of his knowing enough geometry for this feat is as questionable as the possibility of his known enough English to compose the famous palindrome ‘ABLE WAS I ERE I SAW ELBA.” The mathematician Branko Grünbaum investigates further in his paper “Is Napoleon’s Theorem Really Napoleon’s Theorem.” He notes that the result’s first known publication appeared in the 1825 edition of “The Ladies Diary,” four years after Napoleon’s death. Not until 1911 did someone even mention Napoleon alongside the theorem. On through the 20th century, Napoleon’s name became ever more attached to the result. Grünbaum writes that “Napoleon’s theorem” pushes the limits of Stigler’s law of eponymy, which states that “No scientific discovery is named after the original discoverer” (Stigler’s law, in fact, was originated by the sociologist Robert K. Merton). Now, with its common use past the point of likely revision, Napoleon’s name is cemented in the history of geometry.

The 19th century brought the golden age of triangle geometry, with notable points discovered by Gergonne (X7), Nagel (X8), Spieker (X10), Feuerbach (X11), Neuberg (X15,X16), and Brocard (X39,X76). But by the middle of the 20th century, the stature triangle geometry once held had greatly diminished. The rapid progress of the 19th century resulted in the field’s saturation by the 20th, so that, unlike other areas of mathematics, there were in this one no longer any deep unsolved problems. Furthermore, brute-force methods, void of elegance, had sufficed for most problems in the field. A triangle could be converted into equations and bulldozed by calculation. Triangle geometry was no longer worthy of serious scholarly attention. But while triangle geometry was left behind by academia, computers opened the door to further discovery of triangle centers. The field is different now, but it occupies its niche. A devoted group of followers continues to search for “miracles of the triangle.” In a 1995 article memorializing the rise and fall of triangle geometry, the mathematician Philip J. Davis still saw a future for the subject. Invoking Frank Sinatra, he wrote, “The song is ended, but the melody lingers on.”

Between 1999 and 2013, more than 20,000 messages devoted to the esoterica of triangle geometry were exchanged in the yahoo group Hyacinthos. The name, Kimberling says, was established not in honor of Apollo’s lover, but of the French geometer Émile Michel Hyacinthe Lemoine (X6). Among enthusiasts of triangle geometry, Hyacinthos became legendary for its concentration of experts in the field. Long calculations were shared in the group, each discussion seemingly advancing the field’s frontier. Much of the knowledge accumulated in Kimberling’s encyclopedia can be traced to exchanges in Hyacinthos. The celebrated Princeton mathematician John Horton Conway contributed frequently, his messages preserved in archived web pages.

From John Conway
Date: Thursday December 30th, 1999. 5:43pm.
Subject: Re: Re: Fermats and co

I'm continuing to work backwards and will add comments as before:
Choose an angle g and construct "isosceles Napoleons" on the sides of triangle ABC, [vith vertices not on ABC being] A', B', C'.
It is known that AA', BB', CC' concur in a point on the Kiepert hyperbola. As g varies from 0-pi/2, the Kiepert hyperbola is swept out.
... actually g varies modulo pi, so one should take either -pi/2 to pi/2 or 0 to pi.
We can identify the following special cases.
g = pi/3 -pi/3 pi/6 -pi/6 pi/4 -pi/4 0 pi/2
Fn Fs Nn Ns Vn Vs H G
Aha! So Vn and Vs are the "Pythagorean perspectors" in my language.
Can we say that, more generally, the Xn are for g>0 and the Xs for g<0 ?
This convention is OK for the pair of points corresponding to any fixed angle, but care should be taken in using it more generally, because these "n,s" subscripts are a part of my "extraversion" notation, whose properties shouldn't be disturbed.

[…]

I'm sorry - perhaps I introduced that at an irrelevant moment. Anyhow, I'm sure you'll appreciate its value later. [On] intersections of the lines going through the Vn,s with Euler line.
....
B2 = VnFs /\ VsFn is sqrt3
B1 = VnFn /\ VnFn is -sqrt3
A1 = VnIs /\ VsIn is 6+3sqrt3
A2 = VnIn /\ VsIs is 6-3sqrt3
Let me try to work out my suspected generalization:
I'll find PgPh /\ P(-g)P(-h) (writing Pg for P^g)
1
Pg = ( : ----- : ) = ( : (SB-Sg')/bb :)
SB+Sg
in view of the identity I just discussed (I didn't expect to be using it QUITE so soon!). So I'll take the "pro" of your statements, which won't disturb collinearities.

*** I've just realised I'm expected home in 15 minutes, and it takes 20 minutes to walk there! So I'll return to this later. Sorry it's a bit of a mess, but if I think the rest of it out before typing it, that will probably improve things!

John Conway

Conway, whose mathematical interests were notably diverse, had a particular affection for geometry. At a conference years ago, Conway told Kimberling of a notebook he maintained, full of geometry observations dating from his childhood. In the last two decades of his life, Conway worked on a project succinctly titled “The Triangle Book.” His progress slowed when a close collaborator, a high-school teacher named Steve Sigur, died in 2008. With Conway’s death last week, the book remains incomplete. X(384) is the Conway point.

Contribution to triangle geometry continues, and Kimberling keeps the encyclopedia updated. Every day, he expects an email filled with material from Antreas Hatzipolakis (the founder of Hyacinthos), who is Greek, and another from Peter Moses, who is English. Moses’s contributions often come directly copied-and-pasted from the computational software Mathematica and require substantial reformatting for the website. If it weren’t for the high quality of the content, Kimberling says, he might ask Peter to do some formatting beforehand. Randy Hutson, of San Antonio, sends Kimberling an update once a month, which, when printed out comes to forty or fifty pages. César Lozada (Venezuela) sends well-formatted lists of triangle centers. Kimberling asks that proposals for new centers include the points’ coordinates within a triangle of side lengths thirteen, six, and nine. This ensures that the encyclopedia is free of duplicates. About three times a month, Angel Montesdeoco (Spain) sends insightful paragraphs, often with accompanying graphics. Recently, Kimberling has been receiving more and more messages from Brazil, India, and Vietnam. The process for adding contributions can be exhausting—a task of cleaning text and reformatting in HTML. The encyclopedia has its own stylebook, but Kimberling has long since given up on adding the necessary Oxford commas. When I asked him what he enjoys most about triangle geometry, Kimberling, now 77, responded, “Of course, it’s thrilling to ‘discover’ new triangle centers, and conics, and cubics, etc., but at my age, it’s becoming increasingly fulfilling to share the thrills that others have experienced with their own discoveries and developments.” The most recent triangle was discovered by Kadir Altintas and Ercole Suppa as X(37887). It was added on April 15th, 2020. ■

Some notes from June 2026

I wrote this essay in early 2020, during a college course taught by John McPhee. John's editing improved the essay enormously.

In his lovely book Tabula Rasa, he briefly describes the essay (alongside the writing of my classmates):

... Moving on to his next, and less difficult, subject, he encapsulated Euclidean geometry and explained triangle centers to his lay readership, holding to the end the interest of the innumerate (e.g., me). Among thousands of different types of triangle centers, the centroid kind is the most straightforward. Lines drawn from each vertex to the midway point in a triangle's opposite side intersect in one place. No matter the shape of the triangle, or how ungainly it may be, the three lines always cross one another at a single point. As Kenny went to virtual press, the number of triangle centers so far known in the mathematical history of the world was 37,887 and growing.

Six years have passed since then. Clark Kimberling recently told me that "The encyclopedia keeps on growing," relaying to me Peter Moses's recent contribution of "equilateral limit curves." The encyclopedia now includes upwards of 72,000 centers.

I should also mention how I myself became a triangle enthusiast. For that, I must thank my middle school math teacher Mr. Lomas.