ON A THEOREM RELATING TO RECIPROCAL TRIANGLES. 
then the equations of the sides of the triangle are 
A x + B y+G z = 0. 
A' x + B' y + C' z = 0. 
A"x + B"y + G"z = 0, 
and the coordinates of the angles of the reciprocal triangle may be taken to be 
(A, B, C) (A', B', O') (A", B", C")-, the equations of the lines joining the corresponding 
angles of the two triangles are therefore 
(By —C ¡3 ) x 4- (G a — A 7 )y + (A/3 —B a )z = 0, 
(B'y' -C' P')x + (C' a! -A'y')y + (A'p' - B'a ) z = 0, 
(B'Y - G''13") X + (G"a" - A"y") y + (A"(3" ~ B"a") z = 0 ; 
the condition that these lines may meet in a point is therefore 
B y — C ¡3 , C a — Ay , A /3 — B a. 
B' y' - GY’ , Ca! - A'y , A' ¡3' - B' a' 
B'Y - G”(3", G"a" - A'Y, A"!3" - B"a' 
an equation which is satisfied identically when A, B, G\ A', B', (7; A", B", G" are 
replaced by their values. To prove this I transform the different quantities which 
enter into the determinant as follows: putting 
F = 0L'a" + /3'(3" + y'y", 
G = a" a + /3"/3 + y"y , 
H = a. a! + /3 /3' +77'; 
we have 
K (By -C/3) = 7 (y'a." - y"o!) - ¡3 (cr/3" - a''/3') 
= a"(/3/3' + yy') — a! (fifi" + yy" ) 
= a" (aa' + /3/3' + yy) - a' (aa" + (3[3" + 77") 
= oc"H-ol'G, 
&c.; 
and the equation becomes 
a ' H — a! G , ¡3"H - /3' G , y"H-y'G =0. 
a F — a"H, /3 F - /3"H, y F - y"H 
a'G-aF, /3'G-/3F, y'G-yF 
Now the minor {/3F - /3"H) (y'G - 7F) - (yF - y"H) (/3'G - /3F) is equal to 
GH YY - PW) + HF((3"y - py") + FG (Py' - P'y), 
i-e. to K (GHA + HFA' + FGA") ;