SEVENTY-TWO CONSECUTIVE PROPOSITIONS IN TRANSVERSALS. 51 



Let the equation of the line in which P x moves be la + m(3 + ny = ; or sub- 

 stituting the co-ordinates of P x now supposed variable (see Theorem 1), 



/ 9 h 

 0(P x )is fa + g(3 + hy = = u . . . . . (2). 



Now h may be considered as constant, since it is the ratios only that are con- 

 cerned. Take /as the independent variable, then g varies with /by (1) ; and by 

 the theory of envelopes we have 



|=»= ($) + (!)■ I < 3 >- 



Elimination gives =i= (la)* ± (m/3)* rb (ny)* = ; — 



the equation of a conic touching the sides of the triangle of reference. — Q. E. D. 



Theorem LXVI. 



Let lines be drawn from the angles of a triangle through any two points and 

 terminating in the opposite sides ; by joining the extremities of each set of lines 

 so drawn, two other triangles will be formed. The three lines joining the inter- 

 sections of corresponding sides of these two triangles with the corresponding 

 angle of the original triangle meet in a point. 



If the co-ordinates of the two assumed points be a, 5 X c x and a 2 b 2 c 2 those of 

 the third point are 



b v b 2 



C V C 2 



, h b 2 



C V C 2 



a v a 2 



> C l G 2 



Cv-. a Cw-j 



b vh 



Let this point be called the anapole of the two assumed points. 



Theorem LXVII. 



The anapoles of A 7 , A 8 ; of B 7 , P> 8 ; and of C 7 , C 8 are in a straight line, — say #\i IBi ©i 



Theorem LXVIII. 



The straight line joining the anapoles of P x , A 8 and P 2 , A 7 passes through A ; cutting BC (say) in $\ 2 - 



» » >> "l> "8 " 2' ""7 » ■"» » C" » ©2' 



» » » Pi> C 8 „ P 2 , C r „ C; „ AB „ (g 2 . 



^2 ©2 ©2 is a straight line, identical with ^.B 5 © 5 of Theorem XII. 



Theorem LXIX. 



The lines $\i ©i ©n ^ 2 ^ 2 © 2 intersect in the anapole of P x , P 2 , which is also the 

 pole of the line P x P 2 to the imaginary conic, a 2 + /3 2 + y 2 = 0. So that the 



