46 REV. HUGH MARTIN ON TRILINEAR CO-ORDINATES : 



Theorem XX. 



B C, B 7 C 7 , B g C 8 meet in a point, & 9 . 

 CA, C 7 A 7 , C 8 A 8 ,, „ 23 9 . 

 AB, A 7 B 7 , A 8 B 8 „ „ ©,. 



Theorem XXI. 



& 9 33 9 (£ 9 is a straight line. 



Theorem XXII. 



B C, A A 2 , AgPj, A 7 P 2 all meet in a point, & 10 . 

 CA, BB 2 , B 8 P U B 7 P 2 „ „ 33 10 . 



AB, CC 2 , C 8 P X , C 7 P 2 „ „ © 10 . 



Theorem XXIII. 



A & 9 , A C, A iU 10 , A B is a harmonic pencil. 

 B*„ BA, B» 10 , BC „ 



ce 9 , CB, C© 10 , CA „ 



Theorem XXIV. 



B C, A 1 P 2 , A 2 P X meet in a point, & ir 

 CA, B^,, B 2 P t „ „ JJ U . 

 AB, C 1 P 2 , C.jPj ,, „ ffi 11 . 



Theorem XXV. 



A & n , B 33 n , C dt u intersect in a point, P 6 . 



Theorem XXVI. 



B C, AjAg, A 2 A 7 meet in a point, 3 12 . 

 CA, B.Bg, B 2 B 7 „ „ & 12 . 



AB, CjCg, C 2 C 7 „ ,, © 12 . 



Theorem XXVII. 



A & 12 , B 35 12 , C © 12 intersect in a point, Q r 



Theorem XXVIII. 



Let B C, A 7 A 8 meet in & 13 ~J 



CA, B 7 B 8 „ 1S 18 j> ;— Then, A a ig , B B 13 , C© 13 intersect in a point, Q 2 . 



Theorem XXIX. 



— Then, A A 9 , B B 9 , C C 9 intersect in a point, Q 3 . 



Theorem XXX. 



Let AjAg, A 5 A 6 meet in A 10 | 



B X B 2 , B 5 B 6 ., B 10 > ; — Then, A A 10 , B B 10 , C C 10 intersect in a point, Q 4 . 



AB, 



c 7 c 8 



)5 





B 5 B 6» 



C r C r , 



5 6' 



A 5 A 6> 



c 5 c 6 



A 5 A 6 



B 5 B 6 



meet 



in A 9 

 B 9 



c 9 



C X C 2 , C 5 C 6 „ c 



10 



