23(5 ON TRIANGLES. 



Certain proper- angle B C M, 17.1 ; but the angles OKE, A C B are equal 

 ties of triples, (Ru ^ ggA ^ therefore the triang!e M B C is a part of the trian- 

 gle A K C, and is contained in it ; but thefe triangles are as 

 their bafes M B, B A (Euc. 1.6;) confequently M B is apart of 

 BA, Simf. Euc. D. 5; and the point M is between the points 

 A and B. Q. E. D. 



Corollary. Hence it appears that if two lines C M and B N 

 be drawn, and their interferon O lay within the triangle 

 ABC, the place of the third point S will be in the fide B C 

 unproduced. But if O lay without the triangle, the place of 

 the third point will depend on thefitnation of the given points; 

 namely if thefe be in two fides produced, the place fought will 

 lay in the remaining fide unproduced ; but if either of the given 

 points fall between two angles of the triangle, the third will 

 be found in the remaining fide produced. 



Propofition 1. If a right line C M (Fig. 1 .) be drawn from 

 C, one of the angles of a triangle ABC, to cut the oppofite 

 tide A B, produced or not, in M, and if from the point M, 

 perpendiculars, M F, M H be let fall upon the fides A C, C B, 

 the reclangle B M, M F will be to the rectangle A M, M H, 

 as BC is to C A. 



Danonflration. Draw A P, BQ perpendicular to BC, C A; 

 then we have the following proportions, by femilar triangles; as 

 M B : M H : : A B : A P; and as M F : M A : : B Q : B A ; 

 hence as M B x M F : A iMy M H : : B Q : A P. But as B Q : 

 A P : : BC : C A, Euc. 1 t.6; therefore the reftangle M B, 

 M F is to the reftangle MA, MH as BC is to C A. Euc. 

 U.S. Q.E.D. 



Cor. \ft. If O h, 0/ be drawn perpendicular to B C, C A, 

 from any point O in M C; the rerftangle M B, Of will be to 

 M A, O h as B C is to C A; for by fimilar triangles as /O : 

 O Mi ¥ M : M H; henct as M Bx Of: M A x O ft : : M B 

 X M F : M A x M H ; confequently as M B x Of : M A x 

 Oft: ? Be : C A. Euc 11.5. 



Cor. 2nd. Let the triangle ABC be ifofceles, having the 

 fides B A, A C equal; alio let C M, B N, drawn to the op- 

 pofite fides BA, A C meet in O; then the point S is given* 

 in which A O, produced if necefiary, cuts the remaining fide 

 C B. For draw Of, O g perpendicular to C A, A B, and we 

 (hall have, as B S x Of: C S x O g : : B A ; A C, Cqv. 1 ; but 



