398 



then it is easy to see that 



he 



2 cos " 



2 cos J C= fa 

 'Slab 



Substituting the values of y/l, \/ m, n, from these equations in 

 equation (2), it becomes, by dividing by 2*/ 2abc i 



co S Mj| + cos^J| + co S JcJ| = o. (6) 



Now, if S be a circle (fig. 2), and 0 a point whose co-ordinates 

 are substituted in the equation of S, we have, if our equations be Car- 

 tesian, 



$_ _ OP-QQ 

 2a PQ ' 



And when the circle S becomes infinitely large, the limit of OQ : PQ 



is unitv. Hence the limits of — , — , when S 8' S" becomes 



J 2a 2b 2c 



right lines, are the perpendiculars let fall on these lines, and denoting 



them by a, (3, 7, the equation (6) becomes 



cos \ A a + cos I B v//3 +cosJ C ^7 = 0. (7) 



This is the equation of the circle inscribed in the triangle formed by the 

 lines a, /3, 7 ; and the exscribed circles may in like manner be derived 

 from the equations (3), (4), (5). — q. e. d. 



4. The equation of the circle circumscribed about a triangle is also 

 a particular case of equation (2). Thus, let S, S', S", become points, 

 denoting them by A, B, C, and the point S' /f by D, then equation (2) 

 becomes Ptolemy's theorem, 



BC'AD + CA-BD + AB- CD = 0. 



Hence, 



BC CA AB 



BD-CD + CD AD + ADBD ~ ( 8 ) 



