ON TRIANGLES. 235 



the angular points B, A and C, the place of O will be within Certain proper- 

 the triangle, and the three lines CO, BO and A O will fall ties o( tlian S ks » 

 wholly Within (he fame; but if one of thefe points-, as S, lay 

 in BE, which is C B produced, the place of O will be ex- 

 ternal to the triangle; in which cafe, one of the two remain- 

 ing points of interfection will lay in its refpeclive fide pro- 

 duced; but the filuation of the other will always be betwixt 

 the angular points of the third fide. 



Demonfiration. 



Cafe If. Let M, N and S (Fig. 1.) be betwixt the angu- 

 lar points B, A and C; in which cafe, the triangles ABC, 

 M B C are of the fame altitude; confequently they are as 

 their bales A B, B M, Euc. 1.6; but B M is a part of A B by 

 bypolhefis; therefore the triangle M B C is a part of ABC, 

 (Simp fan's Euclid. D. 5.) that is, the triangle ABC contains 

 the triangle MBC; for the fame reafon, the fame ABC con- 

 tains the triangle SAB; but the figure M B S O is common to 

 the triangles M B 0, SAB; therefore it is contained in the 

 triangle ABC; confequently the place of O is within the 

 fame: now as the points A, B, C and O are all found in the 

 fpace ABC, the lines A O, BO and C O mud alio lay wholly 

 within the fame triangle. 



Cafe 'J. When the point S falls in B E or C B produced, 

 (Fig. 2.) the angle C B A is external to the triangle A B S; 

 therefore it is greater than the angle B S A, (Euc. 17, L;J and 

 the fum of the angles CAB, A C B is lefs than the angles 

 C AS, A C S taken together, (Euc. 32. 1); hence the angle 

 CAB is lefs than CAS; confequently the right line A S, 

 produced at pleafure, falls wholly without the triangle A B C. 

 Now if the remaining two lines C M and B N be luppofed to 

 cut the fides B A and A C not produced, the place of O, their 

 common interfection, will be in the fpace ABC, by Cafe I : 

 but this is impoffible; for O is in the right line A S by hypo- 

 thecs; which has been fhewn to fall wholly without the trian- 

 gle ABC, confequently one at leaf! of the lines C M, B N 

 muft alfo lay without the fame fpace; let this be BN, and 

 draw O K parallel to C A ; then the external angle O K E of 

 the triangle O K C is greater than the internal and oppofite 



angle 



