History of the Nine Point Circle

I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. 1

Brewster's Memoirs of Newton. Vol. ii. Chap. xxvii.

Euler was once thought to be the earliest discoverer of the nine point circle. No one, however, has been able to provide evidence or reference to any passage in Euler's writings that support him being the first to make the discovery of the nine point circle. In the Proceedings of the Edinburgh Mathematical Society, MacKay (1892), claims that this attribution is a mistake and explains the origin of the mistake. (see his article for a complete explanation)

You may then ask, "Who is the discoverer of the Nine Point Circle?"

Based on historical research by MacKay (1892), there have been several independent discoverers of the nine point circle. These discoverers cover a wide range of territory, English, French, German, and Swiss.

The following is a listing (timeline) of the publications that refer to the discovery and discussion of the nine point circle.

Historical Development of the Nine Point Circle


Bevan's theorem appears in Leybourn's Mathematical Repository

Proof of Bevan's theorem found in the Mathematical Repository, Vol. I, Part I, page 143 given by John Butterworth of Haggate


John Butterworth poses a question relating to nine-point circle in Gentleman's Mathematical Companion


Two solutions to Butterworth's original question are given in the Gentleman's Mathematical Companion, one given by Butterworth himself, and the other by John Whitley


The nine points are explicitly mentioned in Gergonne's Annales de Mathematiques , volume xi., in an article by Brianchon and Poncelet. This article contains the theorem establishing the characteristic property of the nine point circle.


First enunciation of Feuerbach's Theorem, including the first published proof, appears in Karl Wilhelm Feuerbach's Eigenschaften einiger merkwiirdigen Punkte des geradlinigen Dreiecks, along with many other interesting proofs relating to the nine point circle.


In an article, Symmetrical Properties of Plane Triangles, published in the Philosophical Magazine (1827), II, pages 29-31, T. S. Davies proves the characteristic property of the nine point circle and remarks that the centroid is situated on the line which contains the orthocenter, the nine-point center, and the circumcenter.


In Gergonne's Annales de Mathematiques, xix. pages 37-64, Steiner shows in an article entitled Developpement dune serie de theoremes relatifs aux sections coniques, among other things, that the nine point circle property is only a particular case of a more general theorem.


Steiner publishes his tractate Die geometrischen Constructionen, ausgefuhrt mittelst der geraden Linie und eines festen Kreises where he enunciates the theorem that twelve points associated with the triangle lie on one and the same circle.


The circle is officially designated the "nine point circle" (le cercle des neuf points) by Terquem, one of the editors of the Nouvelles Annales. (see Volume I page 198). Terquem published the second analytical proof of the theorem that the nine point circle touches the incircle and the excircles.

*The nine point circle has also been called the six-points circle, the twelve-points circle, the n-point circle, Feuerbach's circle, Euler's circle, Terquem's circle, il circolo medioscritto, the medioscribed circle, the mid circle, and the circum-midcircle.


J. Mention, in an article entitled Note sur le triangle rectiligne published in Nouvelles Annales, IX, pages 401-403, provides the first geometrical proof of Feuerbach's theorem.


W. H. Levy shows another geometrical proof of Feuerbach's theorem in the Lady's and Gentlemen's Diary, page 56.


T. T. Wilkinson of Burnley publishes work on the nine point circle published in the Lady's and Gentlemen's Diary, pages 67-69.


John Joshua Robinson enunciates and proves a new theorem relating to the nine point circle, published in the Lady's and Gentlemen's Diary, pages 86-89.


Reverend George Salmon calls attention to Feuerbach's theorem in an article dated July 17, 1860, published in the Quaterly Journal of Mathematics, volume IV, pages 152-154 (1861).

In the same volume of the Quarterly Journal (pages 245-252), John Casey has an article, dated November 27, 1860, in which he proves not only Feuerbach's theorem, but also extends the contact to an indefinite number of circles.

MacKay mentions and gives reference to numerous demonstrations of Feuerbach's theorem in his extensive article pertaining to the history of the nine point circle. MacKay goes into great detail about the historical development of the nine point circle; much more than is provided in this brief historical outline.

The following is the complete reference to his well researched and high recommended article:

MacKay, J. S. (1892). History of the Nine Point Circle. Proceedings of the Edinburgh Mathematical Society, (11). pages 19-61.

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