411 



a fifth circle, 2 ; and let G be the centre, and R the radius of 2 ; then 

 we have (fig. 5), 



square of common tangent of S, S r 

 = (R -r)(R- r') ' 



Again, 



Hence, 



OO' 2 



4sm 2 ^AGJB = -^-. 



R 



00' = common tangent of S, S , x , (41) 



*y {R — r) {R — r) 



Now, by Ptolemy's theorem, 



0 O r ■ 0" O 1 " + 0' 0" - 0 0"' +0"0- 0' 0"' = 0. 



Hence, substituting for OO', from (41), and making like substitutions 

 for 0" O'", &c, we have the common tangent of S, S' by the common 

 tangent of S" 9 S'"+ &c, = 0. 



25. The proof given in the last Article is that alluded to in Art. 1 ; 

 and it is evident that it may be proved in a manner precisely similar, 

 if S, S', S", be four circles on the surface of a sphere touching a 

 fifth circle, 2, that the sin J common tangent of S, S' -j- sin common 

 tangent of S", S'"+ sin 1 common tangent of S' f S" x sin ^common tangent 

 of S, S'" + sin -J common tangent of S'\ S x sin J common tangent of 

 S', S /f/ = 0, the common tangents being the direct or the transverse, ac- 

 cording as the contacts of the pairs of circles to which they are drawn 

 with 2 are similar or dissimilar. 



26. The direct application of the theorem in the last two Articles 

 gives at once a proof of Feuerbach's theorem for plane triangles, and of 

 Dr. Hart's extension of it to spherical triangles. 



For if S, S' S", S'", be the inscribed and exscribed circles of a plane 

 triangle, the common tangent of S, S' = h - c j 



of S",S"'=b + c. 



Hence common tangent of S, S' x common tangent of S", S'^-i 2 - & ; 

 and the other rectangles = c 2 - a?, and a 2 - 1 2 , respectively. Hence the 

 condition holds of S, S\ S", S"', being all touched by the same circle. — 



Q. E. D. 



