44 REV. HUGH MARTIN ON TRILINEAR CO-ORDINATES I 



Theorem I. 



A A v B B p C C, intersect, in a point, P r 



Theorem II. 



Let B C, B 1 C 1 meet in S x j 



C A, C 1 A l „ B, J . Then ;—^ l B 1 € 1 is a straight line. 

 AB, A^ „ ©J 



Theorem III. 



C v B t C meet in A 2 ") 



A C. A „ B I. Then ;— A A 2 , B B 2 , C C 2 meet in a point, P„ 



B p A,B „ C 2 J 



Let B C v B t C meet in A 

 C 

 A 



Pj and P 2 are reciprocal points. 



Theorem IV. 



B C, B 1 C p B 2 C 2 intersect in a point, and that point is ^ 

 C A, CjAp C 2 A 2 „ „ )} J3 1 



A B, A^j, A 2 B 2 „ „ „ e t 



Theorem V. 



A X A 2 , BjBj, CjC 2 intersect in a point, P 3 . 



Theorem VI. 



P X P 2 P 3 is a straight line. 



Theorem VII. 



Let A X B, B C meet in & 2 1 And let Cj A, B C meet in & 3 1 



BjC, CA „ 33j; A 1 B, CA „ B 3 



C A, AB „ ©J BjC, AB „ ®J 



Also let B 2 ffi 2 , B 3 e 3 meet in A 3 \ And let 33 2 © 3 , B 3 # 2 meet in A 4 

 © 2 2l 2 , © 3 ® 3 „ B 3 l; 0^, e 3 a 2 „ B 4 



a 2 i3 2 , a 3 B 3 „ c 3 J a 2 B 3 , <a 3 B 2 „ c 4 . 



Then, — A A 3 A 4 is a straight line. 



BB 3 B 4 » » 



C C 3 C 4 „ „ 



Theorem VIII. 



A A 3 A 4 , B B 3 B 4 , C C 3 C 4 meet in a point, P 4 . 



Theorem IX. 



AjA 2 A 3 is a straight line. 



B 1 B 2 B 3 « » 



C 1 C 2 C 3 » » 



