Subject: Re: [HM] Benjamin Bevan's Problem
From: Julio Gonzalez Cabillon (jgc@adinet.com.uy)
Date: Wed Feb 23 2000 - 17:11:44 EST
At 17:22 24/01/00 +0000, Antreas P. Hatzipolakis typed:
|
| <quote>
| Nine-point circle. In any triangle, the mid-points of the sides, the feet
| of the altitudes. and the mid-points of the lines joining the vertices
| to the orthocentre, all lie on a circle.
| Brianchon and Poncelet published the theorem in 1821, though an otherwise
| unknown Englishman, Benjamin Bevan, proposed a problem in 1804 which is
| practically equivalent.
| </quote>
| David Wells: The Penguin Dictionary of Curious and Interesting Geometry.
| Penguin Books, 1991, pp. 158 - 159.
|
| Question: Which was B. Bevan's problem?
|
The Civil Engineer Benjamin Bevan proposed in Leybourn's _Mathematical
Repository_ [ new series, vol I (1804) page 18] the following problem
(without proof):
"In a plane triangle, let O_o be the centre of a circle
passing through I_1, I_2, I_3, then will O O_o = O I,
and be in the same right line, and
O_o I_1 = O_o I_2 = O_o I_3 = 2R
or the diameter of the circumscribing circle."
Best regards from sunny Montevideo,
Julio Gonzalez Cabillon
PS Apologies for my belated reply, but summer is summer in UY!
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