MATHEMATICS: F. MORLEY 
171 
AN EXTENSION OF FEUERBACH'S THEOREM 
By F. Morley 
JOHNS HOPKINS UNIVERSITY 
Received by the Academy, February 17, 1916 
Feuerbach's theorem, that the four circles which touch three lines 
also touch a circle, may be stated thus : given four orthocentric points, 
forming four triangles, the 16 in-circles of these triangles touch the 
circle F on the diagonal points. 
Now each in-circle and the omitted one of the four points is a degen- 
erate curve of class three on the absolute points / /. There is further 
a rational curve of class three, on the six joins of the four points and 
touching the Hne infinity at / J, which touches F three times. Thus the 
theorem is suggested: All circular line-cubics on the joins of four ortho- 
centric points touch the Feuerbach circle. 
A proof is as follows. It is convenient to state the algebra dually. 
That is, we have 4 Hues 1, =t= 1, =t 1 and a pair of lines, ^ and 1/^, apo- 
lar to all conies on the 4 lines. Two point-cubics on the six joins of the 
4 lines meet again at 3 points xy z, which are points of contact of a 
tri tangent conic of either cubic. When x and y are given, z is ration- 
ally known; and when x is given and z moves on a line ^, we know from 
the theory of the Geiser transformation that y moves on a rational 
quartic px^ which has a triple point at x. There is then a connex of 
the form 
where z is on f . And if ^ be the join of x and y this connex is of the 
form 
^ e^cy. 
(1) 
If ^ be 1, 0, 0, the quartic in y is the two lines 
Xo Xi X2 
yo yi yi 
0 1 =1= 1 
and the conic on x and the 4 other points, that is 
X^ — X2^, Xi X2 
yi' - y2\ yxy% 
Hence the connex is explicitly 
2 f 0 (^1^ - ^2^) { (x^yx -f x^y^ ^, - (a:o>'2 + ^2^0) ^2 
H- 2 {x^y^ - x^y^ U \ =0. 
(2) 
