46 PHILOSOPHICAL SOCIETY OF WASHINGTON. . 
were first published in January, 1821, in Gergonnes’ Annales, vol. 
II, in a memoir by Brianchon and Poncelet on the determination 
of an equilateral hyperbola from four given conditions. 
In 1822 Prof. K. W. Feuerbach, of Nurenburg, showed that this 
circle is tangent to the inscribed and three escribed circles. This 
property is known as Feuerbach’s theorem, and the circle is known 
to the Germans as Feuerbach’s circle, In 1828 Steiner showed that 
this circle passed through twelve noteworthy points and was tangent 
to the in and escribed circles. This was done without a knowledge 
of the earlier work by Feuerbach. 
In 1842 Terquem, the editor of the Nowvelles Annales de Mathé- 
matiques, called it the nine-points circle. In some books it has been 
called the six-points circle. In an article, by myself, in the Mathe- 
matical Magazine for January, 1882, I have called it the twelve- 
points circle. 
Of the twelve points considered noteworthy six are in the sides 
of the triangle and six are not. If all the noteworthy points now 
known are to determine its designation, then twelve-points circle ap- 
pears to be a proper designation. If only those points in the sides 
of the triangle should determine the designation then six-points circle 
would appropriately name it. In either case nine-points circle would 
be an imperfect designation, and as the name Feuerbach’s circle was 
the first name it received it seems on the whole best to adhere to it. 
A somewhat analogous case is the seven-points circle known as Bro- 
card’s circle, not named for the number of noteworthy points it 
contains but for its discoverer. The name twelve-points circle may 
therefore be rejected and the name Feurbach’s circle adopted. 
The following proof of the fundamental properties of Feuerbach’s 
circle is offered as being simpler than that usually given: 
PRELIMINARY. 
1. Definition. When we determine upon a right line AB a point 
Csuch that AC = BC = 2 AB the line is said to be bisected. 
This restricts us to one point C. 
Fig oy A C B Ge 
I t ! I 
It will be found convenient in what follows to extend this 
definition so as to include the points C’ and C”, Fig. 1, where 
AC= BC= AC’ = BC” = 3 AB; 1. ¢., by considering a der 
