Re: [HM] Six point Circle


Subject: Re: [HM] Six point Circle
From: Ken Pledger (Ken.Pledger@vuw.ac.nz)
Date: Mon Jun 19 2000 - 18:12:52 EDT


>
> Is it true that Euler (1765) had a six point circle:
>
> The centers of the three sides of a triangle, and
>
> the footpoints of the three heights.
>

        The question of why Euler's name is ever attached to the 9-point
circle came up recently in the Hyacinthos mailing list. It may be helpful
to repeat here the message which I posted about it.

> The attribution to Euler seems to be wrong. I've recently received
> a copy of a very thorough discussion by J.S. Mackay, "History of the
> Nine-point Circle," Proc. Edinburgh Math. Soc. XI (1893), 19-57. Mackay
> begins as follows.
>
>> The earliest author to whom the discovery of the nine-point circle
>> has been attributed is Euler, but no one has ever given a reference to
>> any passage in Euler's writings where the characteristic property of this
>> circle is either stated or implied. The attribution to Euler is simply a
>> mistake, and the origin of the mistake may, I think, be explained. It is
>> worth while doing this, in order that subsequent investigators may be
>> spared the labour and chagrin of a fruitless search through Euler's
>> numerous writings.
>> Catalan in his "Theoremes et Problemes de Geometrie Elementaire",
>> 5th ed. p.126 (1872), or 6th ed. p.170 (1879), says that the learned
>> Terquem attributed the theorem of the nine-point circle to Euler, and
>> refers to the "Nouvelles Annales de Mathematiques", I. 196 (1842). If
>> the first volume of the "Nouvelles Annales" be consulted, it will be
>> found that Terquem has two articles on the rectilineal triangle. The
>> first (pp. 79-87) is entitled "Considerations sur le triangle rectiligne,
>> d'apres Euler"; the second (pp. 196-200) has the same title, but "d'apres
>> Euler" is omitted. In the first article Terquem mentions that Euler
>> discovered certain properties of the triangle, and refers to the place
>> where they are to be found ("Novi Commentarii Academiae ...
>> Petropolitanae", xi. 114, 1765). He says he thinks it useful to
>> reproduce them with some developments, and this is exactly what he does,
>> for the first article is a synopsis of Euler's results, and the second
>> article, which begins with the property of the nine-point circle,
>> contains the developments.
>> Who, then, is the discoverer of the nine-point circle?......
>
> The story then becomes complicated, but perhaps at least I've typed
> enough of Mackay's paper to explain away the Euler myth.

        Paul Yiu followed up with this:

> .... this paragraph of MacKay's .... has confirmed
> my impression that Euler did not write explicitly on the nine-point
> circle. Euler's plane geometry papers are collected in volumes 26 -- 29
> of his Opera Omnia, series prima, and it appears that the most natural
> place he would have written about the nine-point circle is
>
> 325. Solutio facillis problematum quorundam
> geometricorum difficillimorum, Nova commentarii
> academiae scientiarum Petropolitanae 11, (1765 -1767)
> pp. 103--123. I26, 139--157.
>
> In his paper, Euler investigated the problem of
> constructing a triangle given O, I, H. In so doing
> he developed many of basic theorem formulae in triangle
> geometry, including the Euler line. But the nine-point
> circle did not appear here and in other shorter papers
> in the ``geometry volumes''.
>
> Best regards.
> Sincerely,
> Paul Yiu

        Perhaps now some HM subscribers may be able to help lay to rest the
widespread myth about Euler and the 9-point circle.

                Ken Pledger.



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