Subject: [HM] Heron formula and the excircles (3)
From: Paul Yiu (yiu@fau.edu)
Date: Mon Feb 14 2000 - 09:59:05 EST
Dear friends,
(continuing from (2))
It is certainly reasonable for Feuerbach, to make
his treatise self contained, to begin by detailing
the basic formulae for the incircle and the excircles.
But it is striking that most history books [that I have],
when talking about the Feuerbach theorem, give
disproportionately detailed description of the escribed
circles.
***********
For example, Morris Kline seems to focus on the
existence of the nine-point circle, saying that
it was
<quote>
first published by Gergonne and Poncelet.
It is often credited to Karl Wilhelm Feuerbach ...
who published his proof in Eigenschaften ...
<quote>
Then Kline continued,
<quote>
In this book, Feuerbach added another fact about
the nine-point circle. An excircle (escribed circle)
is one which is tangent to one of the sides and to the
extensions of the other two sides. (The center of an
escribed circle lies on the bisectors of two exterior
angles and the remote interior angle). Feuerbach's
theorem states that the nine-point circle is tangent
to the inscribed circle and the three excircles.
<quote>
I am a little bit surprised by a lack of excitement
of the Feuerbach theorem. Does Kline's unnecessarily
detailed description of the excircles here indicate that
the excircles were relatively new in Feuerbach's time?
********
I cannot find the name of Feuerbach or even the
nine - point circle in Smith's two volumes of History.
Eves gave the Feuerbach theorem as an exercise in his
book. The newer ones, Katz and Burton, do not mention
it at all, nor does the one by Grattan-Guinness that
came out recently.
*********
Referring to the Feuerbach theorem, Merzbach, in her revision
of Boyer's book (2nd ed., p.537) wrote:
<quote>
One enthusiast, the American geometer Julian Lowell
Coolidge (1873 -- 1954), called this "the most
beautiful theorem in elementary geometry that has been
discovered since the time of Euclid". * It should be
noted that the charm of such theorems supported considerable
investigation in the geometry of triangles and circles
throughout the nineteenth century.
<quote>
* Coolidge, J.L., "The Heroic Age of Geometry", Bull.
Amer. Math. Soc. 35 (1929) 19 -- 37.
Merzbach began this paragraph with
<quote>
[t]he history of geometry in the nineteenth century is replete
with cases of independent discovery and rediscovery.
<quote>
Then she referred to the publication of the nine-point circle by
Brianchon and Poncelet in 1821.
**************
Boyer, however, put it this way in the first edition.
After mentioning Brianchon and Poncelet on p.572, he wrote
<quote>
That the nine-point circle should be known as Feuerbach's
circle is justified not by priority of publication, but through
other properties disclosed by Feuerbach. The center of the
nine-point circle, he showed, lies on the Euler line and is
midway between the orthocenter and the circumcenter. More
remarkable still is the property contained in what is now known
as Feuerbach's theorem: the nine-point circle of any triangle is
tangent internally to the inscribed circle and tangent externally
to the three escribed circles, possibly "the most beautiful
theorem in elementary geometry that has been discovered since
the time of Euclid."
<quote>
The closing sentence of this paragraph, however, seems to me
an anticlimax:
<quote>
In connection with the proof Feuerbach made use of the fact
that the radii of the inscribed and escribed circles are given
by
(2K)/((+/-)a (+/-)b (+/-)c)
where not more than one minus sign is used.
<quote>
I am somewhat puzzled by the excessive description of the excircles
and the quotations of the formula for the exradius these authors gave
in connection with the Feuerbach theorem.
To be continued.
Sincerely,
Paul Yiu
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