399 



Now, BC, CA, AB, are proportional to sin A, sin B, sin C; and if 

 the equations of the lines BC, CA, AB, be a, /3, 7, we have 



BD ' CD = a ' diameter of circle ; 

 CD- A 0 = p - diameter of circle ; 

 AD BD = 7 • diameter of circle. 



Hence equation (8) becomes 



sin A sin B sin C „ 



+— 3-' + =0. (9) 



a p 7 > ' 



This is the equation of the circumscribed circle. 



5. The equation of the inscribed circle of a plane triangle has been 

 derived in Art. 3 from the equation (2) of a pair of circles touching 

 three circles. Conversely, the equation of a pair of circles touching 

 three others may be derived from the equation of the inscribed circle of 

 a plane triangle. 



For let 2 (fig. 3) be the circle inscribed in the triangle ABC, the 

 equations of whose sides are a = 0 ; (3 = 0 ; 7 = 0; and let the circles 

 S, S f , S", touch 2 at its points of contact with the sides of the triangle 

 ABC; then denoting the radius of 2 by R, and the radii of S, S', S", 

 by r, r, r", respectively ; if any point Q be taken in 2, the result of 

 substituting the co-ordinates of Q in $= 2(E - r) multiplied by the re- 

 sult of substituting the co-ordinates Q in a. 



This may be written 



in like manner, 



S 



and 



2{R-rY 

 S' 



2{R-r') ; 

 8" 



7 2(R-r")' 



Again, since 1* denotes the direct common tangent to S\ S", it is 

 easy to see that 



J(R-r 



I 



cos iA 



(R-r'){R-r")' 



