238 



ON TRIANGLES, 



Certain proper- fite fides B A, A C and C B, in the points M, N and S ; If 

 ties ot truncles* 



A N be to N C as A P B is to B P C ; but B M be to M A as 



B r C is toC r A; (hen will BS be to S C as the produd of 



B P A into B r C is to the produd of B P C into A r iVI. 



Demonftratwn. As A N : N C : : B P A : B P C ; and as B M 

 : M A : i B'C : OA by hypothefis ; hence as B M x A N : 

 CNxAM:: B P A x B r C : B P C x C r A ; but as B M x A N : 

 N x A M : : B S : S C ; confequently B S is to S C as the 

 produd of B P A info B r C is to the produd of B P C into C r A. 

 Buk. II. 5. Q. E. D. 



Corollary ]Jl. The preceding theorem is general ; for the 

 latios of B M to M A, and of A N to N C can be expreded 

 by fome power or powers of the lines B C, C A and A B, 

 BC, which contain the angles oppollle to the fides B A and 

 A C ; therefore if the points M, N and S be arranged accord- 

 ing to the corollary to Proportion 1ft, and the fegments be- 

 twixt thefe points and the adjacent angles have the ratios al- 

 igned in the pre font theorem, right lines, drawn from M, N 

 and S to the oppofite angles, will all meet in one point. 



Cor. 2. When r and p are equal, B S is to S C as B P A is 

 to A P C by the proportion ; therefore if the fegments betwixt 

 the points M, N, S and the adjacent angles are in the ratios of 

 the adjacent tides raifed to any given power, right lines, drawn 

 from M, N and S to the oppofite angles, will all meet in one 

 point by Cor. \jl. 



Cor. 3rd. Let p = o ; then as B S : S C : : B°A : A°C 

 by the proportion; but B°A : A°C is the ratio of equality ; 

 therefore if M, N and S bifed the three fides of a triangle, 

 lines drawn from (hem to the oppofite angles will meet in one 

 point by Cor. 2nd. 



Cor. 4>, help = 1 ; alfo let M and N interfed the fides 

 B A, A C not produced; then S lays between the angular. 

 points B, C, Cor. Pro;). 1 ; and as B S : S C : : B A : A C, 

 Cor. 2 ; therefore the right line S A bifeds the angle B A C ; 

 confequently if three right lines bifed the three internal an- 

 gles of a triangle, they (hall all meet in one point by the laft 

 propofition. 



Cor. 5. Let p = 1 as before ; alfo fuppofe M to be in the 



fide B A not produced; but let N be witlnrat the triangle ; 



then S is in C B produced, Cor. Prop. 1 ; but as B S : S C : : 



5 BA 



