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ROYAL SOCIETY OF CANADA 



Every triangle divides the plane into two regions. There is the 



region which lies on the A side of a, the B side of b, and the C side 



of c, which we may speak 

 of as the region within the 

 triangle ; and there is the 

 rest of the plane constituting 

 the other region. Evidently 

 in passing from one of these 

 regions to the other we pass 

 from one side to the other 

 of at least one of the straight 

 lines a, b, c; and therefore 

 our sect-train has a point 

 in common with one of 

 the sects A B, B C or C A 

 [Thm. 7]. 

 Theorem S. — There are straight lines wholly without a triangle 



and in its plane; and no straight line is wholly within a triangle. 

 For the St' aight line determined 



by the points 1) and E cannot again 



meet a or b. Nor can it meet A B 



for then [11, (4)] it would meet A C 



or B C, i.e., would again tneet b or a. 

 Again, any straight line (^) through 



L, a point within the triangle, has 



also a point on the side. For another 



line through L, and also through M, 



has a point on another side, say N 



on B C [11, (4)]. Hence I, through a point on MN, a side of the triangle 



M N C, must also have a point in common with either C M or C N, i.e., 



with A C or B C [II, (4)]. Hence, too, cutting A C or B C, it must 



[II, (4)] cut another side also. 



III. The assumptions of congruence. 



We might, in a measure, describe conditions here by saying that 

 the fundamental principle is uniqueness, — the principle of there-isonly- 

 one-such. 



(1). If A, B be points on a straight Hne a, and A' a point on straight 

 line a'. Then on a', on one of the two rays from A', we can find only 

 one point B', such that the sect A' B' is congruent to sect A B. Written 

 AB=A'B'. 



