SEVENTY-TWO CONSECUTIVE PROPOSITIONS IN TRANSVERSALS. 45 



Theorem X. 



BC, B^, B 2 C„ B 3 C 3 all meet in & r 

 CA, C X A 15 C 2 A 2 ; C3A3 „ 33 r 



AB, A^ v A 2 B 2 , A3B3 „ e v 



Theorem ' XI. 



LetB C, B 4 C 4 meet in &A 



C A, C 4 A 4 „ 2S 4 } . Then ;— & 4 1S 4 © 4 is a straight line. 

 AB, A 4 B 4 „ ej 



Theorem XII. 



Let B^, A X B meet in A 5 ^ And let B C p C A x meet in A 6 } 



C X A, B X C „ bA;— CA X , A B x „ B 6 \. 



A t B, C X A „ C 5 J AB 15 BC X „ C 6 J 



; B C, A 5 A e 

 CA, B 5 B 6 

 AB, C 5 C 6 



Also let B C, A 5 A 6 meet in 



C A, B 5 B 6 „ 2Sg }■ . Then ;—& 5 3$ 5 © 5 is a straight line. 



Theorem XIII. 



B C, B 5 C 6 , B 6 C 5 meet in a point, & 6 . 

 CA, C 5 A 6) C 6 A 5 „ „ 2S 6 . 

 AB, A 5 B 6 , A 6 B 5 „ „ ® 6 . 



Theorem XIV. 



& 6 3S 6 ® 6 is a straight line; reciprocal to ^ X 33 1 @ 1 . 



Theorem XV. 



Let B C, t3 x © 6 meet in % \ 



C A, ©, a 6 „ 33 7 L Then ;— & 7 25 7 @ 7 is a straight line. 

 AB, a^e „ ©J 



Theorem XVI. 



Let B C, 33 6 © x meet in & 8 ) 



C A, @ 6 gL „ ®A. Then ;— & 8 B 8 ® 8 is a straight line ; reciprocal to & 7 2S 7 ©» 



ab, a,^ „ ej 



Theorem XVII. 



A A v BB r , C C 6 meet in a point, A 7 . 

 BB X , CC 5 J , AA 6 „ „ B 7 . 

 CC X , AA 5 , BB 6 „ „ C 7 . 



Theorem XVIII. 



A A 2 , B B 6 , C C 5 meet in a point, A 8 . 

 BB 2 , CC 6 , AA 5 „ „ B 8 . 

 CC 2 ,AA 6 , BB 5 „ „ C 8 . 



Theorem XIX. 



A 7 A 8 , B 7 B g , C 7 C 8 meet in a point P 5 . 

 VOL. XXIV. PART I. N 



