SEVENTY-TWO CONSECUTIVE PROPOSITIONS IN TRANSVERSALS. 



Theorem 14. 



43 



fl is 



o, 



h+fg, 



-(9+V) 



« 1S 



-( h +M 



o, 



f+gh 



6 is 



9 + V> 



-(f+gh), 







= 0. Therefore ;— 



a 6 23 6 ® 6 is a straight line, viz., f+gh, g + hf, h+fg. 

 ^IjBj^ and & 6 i3 e @ 6 are inverse or reciprocal lines. 



t'X 



is 



is 



Theorem 15. 



(h+fg) -(f+gh), (g+hf). (h+fg), (f+ghf 



(g+¥)\ (f+gh).(g+¥), (h+fg). (f+gh) 

 (f+gh).(g+¥), (h+fgf, (g+¥).(h+f g ) 



and meets B C in & 7 

 CAiniS 7 

 A B in ©„ 



o, (f+gh) 2 , 



-(h+fg). (f+gh), o, 



(h+fgf, -(f+gh).(g+hf), 



(g+¥).(h+f g ) 

 (g+¥f 







= 0. Therefore ;- 



l 7 !5A ^ a straight line, viz., f±& ^-, ^±g- 



riSg^ meeting BC in 

 Similarly, ^ @ 6 ^ „ 



f i 



*6*"i 



C A in 33 8 

 A B in O 



Theorem 16. 



8 ® 8 is a straight line, viz., 



h+fg f+gh g + hf 

 g + hf h+fy' f+gh' 



& 7 3S 7 ® 7 and H 8 H$ 8 <£ 8 are reciprocal lines. 



Theorem 17. 



AAj is 

 B B 5 is 

 C C„is 



o, 



g, - h 



1, 



0, hf 



1,- 



-fg, o 



0. Also 



' A A 2 is 

 BB 6 is 

 C C 5 is 



o, 



h, 



-g 



hf, 



o, 



l 



fg, 



-i, 







= 0. Therefore : — 



A A x , B B 5 , C C 6 meet in a point A 7 viz., fgh, h, g. 



BB X> CC 5) AA 6 

 CC 1; AA 5 ,BB 6 



B, 



h, fgh, f. 



g, /> fgh- 



Theorem 18. 



AA 2 , BBg, C C 5 meet in a point A 8 viz., I, fg, hf. 

 BB„, CCA A. „ „ B 8 „ fg, 1, gh. 



C 8 „ hf, gh, 1. 



BB 2 , CC 6 ,AA 5 

 CC 2 , AA 6 , BB 5 



But instead of continuing the manipulation, we shall gather up these results, 

 and continue the series of propositions. 



