[159 
D 
160] 
egral 
= b give 
160. 
ON A THEOREM RELATING TO RECIPROCAL TRIANGLES. 
[From the Quarterly Mathematical Journal, vol. i. (1857), pp. 7—10.] 
The following theorem is, I assume, known; but the analytical demonstration of 
trgument it depends upon a formula in determinants which is not without interest. The theorem 
referred to may be thus stated: 
“ A triangle and its reciprocal are in perspective; ” where by the reciprocal of a 
triangle is meant the triangle the sides of which are the polars of the angles of the 
first-mentioned triangle with respect to a conic; and triangles are in perspective when 
the three lines forming the corresponding angles meet in a point, or what is the same 
thing, when the three points of intersection of the corresponding sides lie in a line. 
Let the equation of the conic be 
£' 2 -f y 2 + Z 2 = 0, 
and take (a, ¡3, y), (a', /3', y), (a", ¡3", y") for the coordinates of the angles of the 
triangle, then if K be the determinant, and (A, B, C) (AB', O') (A", B", C") the 
inverse system, i.e. if 
KA = (J3'y"-/3”y'), KB = y a!' — y"a!, KG = a'¡3" - a"(3’, 
KA' = (/3"y — ¡3 y"), KB' = y"a — 7 a”, KG' — a"¡3 — a /3", 
KA" = (/3 y — ¡3' 7 ), KB" = 7 o! — y a , KG" = a ¡3' — a' ¡3 , 
equations which may be represented in the notation of matrices by the single equation 
a , /3 , 7 
-1 = 
A, A', A" 
J, y 
B, B\ B" 
a", /3", y" 
G, G\ G"