50 REV. HUGH MARTIN ON TRILINEAR CO-ORDINATES : 



Theorem LIX. 



Let P X Q 6 and P 2 Q 2 meet in TJ 1 "1 T , 



P^andP^Q, „ U 2 ) inen; ~ 



UjUg lr is a straight line. 



Theorem LX. 



V x V a ; R^, R 3 R 4 ; S X S 2 . S 3 S 4 ; Q^, Q 4 Q., Q 2 Q ; all intersect in P. 



Theorem LXI. 



AT is the pole of P^PgP^.Pg to the imaginary conic, a 2 + fi 2 + y 2 = 0. 



Theorem LXII. 



The lines AA X , BB 1? CC X ; AA 2 , BB 2 , CC 2 cut the sides of the triangle ABC 

 in six points which lie in the conic ; — 



*+p+i>- (i+l)f>y- C+{)7«- (£+$)•*-«■ 



For the six points are inverse or reciprocal points. Substituting the co-ordinates 

 of five of them (the third of the second set being omitted) in the general equation 

 of the second degree, and eliminating the arbitrary constants, gives the conic as 

 above. To substitute the co-ordinates of five of the points, omitting now the 

 third of the Jirst set, amounts evidently to inverting the separate terms in the 

 constants of the above equation ; and as this leaves it unaltered, the proof of the 

 theorem is obvious. 



Theorem LXIII. 



The lines AA 5 , BB 5 , CC 5 ; AA e , BB 6 , CC 6 cut the sides of the triangle ABC in 

 six points which lie in the conic ; — 



«. + * + 7 .- (* + i)07- (''/+^)7«-(/?+!)«£ = 0. 

 Theorem LXIV. 



The points 8^3$^ ; ^ 2 15 2 € , 2 lie in the conic;— 



Theorem LXV. 



If the point P x move in a straight line, the line (P x ) will always touch a 

 conic which touches the three sides of the triangle ABC. 



