ON TRIANGLES. 237 



B A, A C are equal by hypothefis; therefore B Sx Of — C S Certain proper- 

 xOg; hence as O g : Of: : B S : S C ; confequently the ties of trian8le5 ' 

 point S may be found, proper attention being paid to Cor. 

 Prop. I. 



Proportion 3. If three right lines C M, B N, A S, drawn 

 from the three angles of a triangle ABC ( Fig. 1 and 2) meet 

 each other in a point O, and the oppofite fides B A, A C and 

 C B, in M, N and S; the fegments of any one fide are as the 

 rectangle under the alternate fegments of the remaining two 

 fides; i. e. B S is (o S C as the redlangle BM, A N to the 

 rectangle C N, A M. 



Demonftrution. Draw O/, O g, Oh perpendicular to C A, 

 ABandBC. Then as BC : CA:: BMxO/: AMxOA; 

 Cor. l. Prop. 2 ; for the fame reafon asAB:BC::NAx 

 O h : N C x O g; hence as B A : A C : : M B x N A x 0/ : 

 NCxMAxbV; butasB A : AC :: ESxOf: CSxO^; 

 confequently B S is to S C as the reclangle B M, A N is to the 

 redangle C N, A M. Q. E. D. 



Cor. If two right lines C M, B N be drawn from two an- 

 gles of any plain triangle ABC, fo as to meet each other in 

 a point O, and the oppofite fides B A, A C in M and N; a 

 point S may be found in the remaining fide, produced if ne- 

 ceflary, fo that if S A be joined, the right line A S (hall pals 

 through O. For fince the right lines B M, MA, AN and 

 N C are given, the rectangles B M, AN and C N, A M are 

 alfo given ; confequently the ratio of thefe rectangles is given : 

 but the right line A O cuts the fide C B, produced or not, ac- 

 cording to circumftances, in the fame ratio, by the laft: pro- 

 pofition ; therefore if both M and N lay betwixt the angular 

 points of the triangle, or in their refpeclive (ides produced, 

 divide B C in S, fo that B S may be to S C in the ratio of the 

 rectangles B M, A N and C N, A M ; then will S be the re^ 

 quired point by Cor. to Prop. 1 ; but if M befiluated betwixt 

 the angles B, A, and N lay in C A produced ; find S in C B 

 alfo produced, fo that B S may be to S C as the reclangle B M, 

 AN is to the re&angle C N", A M ; and S will be the re- 

 quired point, by [he corollary laft referred to. 



Prop. 4. Let three right lines C M, B N and A S (Fig. 1 

 and 2) drawn » >iti the' three angles of a triangle ABC, in- 

 tellect each oilier in a common point O, and meet the oppo- 

 fite 



