48 REV. HUGH MARTIN ON TRILINEAR CO-ORDINATES : 



Theorem XXXIX. 



Let a x /3 2 , y 1 a 2 meet in a A And let a 8 ft, y 2 a 1 meet in a 



ft 7 8 . gift » ft>; p 9 y v a 2 /5 x n fl 



7i« 2 >ft7 2 » 7 4 J 7 2 «i»ft7i » 7 



A a.a, is a straight line. 

 Then;- B ft/3 3 „ „ 



C 7*7 5 



Theorem XL. 



A a 4 a 5 , B /5 4 /3., C 7 4 y 5 meet in a point. 



Theorem XLI. 



Postulating again, similarly as in Theorems XXXVI. and XXXVII.,- 



Let A A 2 , B 7 C 7 meet in L ; and similarly "» 

 . , , A _ „ - \ . Then ; — 



Also A Aj, B 8 C 8 ,, ? 3 ; and similarly J 



a \ ft 7i ^ s a straight line. 



Theorem XLII. 



a 2 ft y 2 is a straight line. 



Theorem XLIII. 



a 3 ft 73 * s a straight line. 



Theorem XLIV. 



A a 4 a 5 is a straight line. 

 Bftft „ „ 

 c 7 4 7 5 



>) )! 



Theorem XLV. 



A a 4 a 5 , B /3 4 (3 5 , C Y 4 y 5 meet in a point. 



Theorem XLVI. 



Let a y a 2 , a 1 a 2 meet in a 6 



Aft, ft ft „ ft Then;- 



7i7 3 . 7i7 2 » 7 a 



A a 6 , Bj9 6 , C7 6 ; A 7 A S , B 7 B 8 , C 7 C 8 all intersect in P 5 . 



Let P 3 Q 3 and P 6 Q 1 meet in Q 5 ; and let P 5 Q 5 and P 4 Q 4 meet in Q 6 . 

 Then the points P 15 P 2 , P 8 , P 4 , P 5 , P 6 ; Q 1; Q 2) Q g> Q 4> Q., Q g have very remarkable relations. 



Theorem XLVII. 



Pj, P 2 , P 3 , P 4 , P 5 , P 6 all range in a straight line. 



