40 



REV. HUGH MARTIN ON TRILINEAR CO-ORDINATES 



Hence the trilinear co-ordinates of A x are 



of B x are 



-1, 



sin (C + Cj) 



sin C x 



sin (C + CJ sin (B + B ,) 

 sin C 2 sin B y 



sin (A+AJ 



sin A, 



of C x are 



sin f B + B,) sin (C + C,) 

 sin B x sin C t 



-1. 



sin(A + A.) sin(B + B.) sin (C + C.) u _. . m . 

 For ;• T > .. I * » . I , P^/, g,A. Then;— 



AAj is 

 BBj is 

 CC X is 



sin A 



sin B 



sin C, 



A x is — 1 



h, g.^ r A is 1, 0, 0. ^ 



~B[ is h, -1, /.I. Also, -J B is 0, 1, 0. I Hence ;- 

 C^s g, f -l.J (Cis 0, 0, l.J 



0, g, -h. 



-/, 0, h. 



/, ~9, 0. 



= 0. Therefore A A v B B p C C l intersect in a point — say P,. 

 V 1 is gh, hf,fg, or f~\ g~\ h~\ 



Theorem 2. 



B^ is 1-f, h+fg, g + hf 

 C 1 A 1 is h+fg, l-g 2 , f+gh 

 A^j is, g + hf f+gh, 1-A 2 



Also, 



BC is 1, 0, 0. 



CAis 0, 1, 0. 

 AB is 0, 0, 1. 



B C, B 1 C 1 meet in 2^ which is 

 CA, C 1 A l „ 3S X „ 



AB, A^ 



0, 9 + V, -(>>+/</)• 



-(f+9fy 0, A+/S'- 



Hence ; - 



= 0. Therefore ;- 



l^^ is a straight line, viz., (f+gh)- 1 , (g + hf)~\ (h+fg)- 1 . 



BCjisl, 0, g 

 CA^sA, 1, 

 ABjisO, / 1 



AA 



is 



B 



B, 



is 



C 



c 2 



is 



Theorem 3. 



Also 



B^is 1, h, 



(1LC 



(AjBis^, 0, 1 



is 0, 1,/V. Hence^ CA^ C t A 



B C v B t C meet in A 2 , viz. 



AB p AjB 



B 2 , „ 

 C 2 , " 



~ h 9, 9* h 

 /, ~hf h 



/> 9- -f'J 



h 

 ~9 



0, -K g 



0, -/ 

 /» o 



= 0. Therefore A A 2 , B B 2 , C C 2 intersect in a point — say P.,. 



P 2 is /, g, h. 

 P x and P 2 may he called inverse or reciprocal points. 



BCis 

 B 1 C 1 is 

 B 2 C 2 is 



Theorem 4. 

 l, o, o. 



l-/ 2 , h+fg, g + hf 



s*(W f ), fQ>+f9), f(9+¥)- 



B C, B 1 C X , B 2 C intersect in % v 



* CA, C^, C 2 A 2 „ iS r 



AB, A^, A 2 B 2 „ © r 



0. Therefore 



By symmetry. 



