'320 Transactions. 



If t lt U, t 8 be the lengths of the tangents to the circle V from A, B, C 

 respectively, then X x = t x 2 + p 2 , Y 1 = U 2 -f p' 2 , Z = tr 4- p' 2 , and the 

 equation of the circle takes the form 



V= (t 2 aa + t.?b(3 + t s 2 c y ) (aa + bfi + c y ) - abc S = o. 



If V touch the circle ABC, then expressing that the radical axis of the 

 two circles is a tangent to S we have 



at, + bt 2 + ct, = o, 



from which we obtain Ptolemy's theorem if we suppose V to reduce 

 to a point-circle. 



This extension of Ptolemy's theorem may be proved geometrically as 

 follows : — 



Suppose the circle V to touch the circle ABC' at the point O on 

 the arc BC, and let AO, BO, CO meet V in the points D, E, F respec- 



OP OR 



tively ; then Uj == BO . BE and t 2 == CO . CF. Also ~ = ^, and 



O r O C 



"D"tl AT) 



therefore — = — , hence t 2 : t 2 = OB' 2 : OC 2 ; 

 CF OC 



i.e., t x : t., : t s = OA : OB : OC. 



By Ptolemy's theorem BC . OA = AC . OB + AB . OC ; 

 i.e., at-i = bt^ + ct A . 



If t be the length of the tangent to the circle V = o from any point P 

 ("cft,yo). th en 



2 a t 2 = aa t 2 + b/3 Q t 2 + cy t? - 2RS,,. 



For the circle BPC, t = t, = t s = o. 



Consider now the circles BPC, CPA, APB. We have 

 aaJS = b{3JS = cyjy = 2ES . 



Hence """ + 6 & + c 7o __ 1 , 1 . 1 

 HenCe ^RS^ t* + tf + t 2 



111 1 



l - e > n + ri + 7i 



t 2 t£ ' t, 2 R' 2 - a 1 ' 2 

 where a 7 is the distance of P from the circum-centre of the triangle ABC. 



If P be the symmedian point of the triangle ABC. then at x = bt 2 = ct s : 

 if G be the centroid of the triangle L 2 = t? — t 2 — ^ 2 (a 2 ). 



If the circle V reduce to a point Pj whose tripolar co-ordinates are 

 (XjYjZi) we have 



X,«a + Y,bf3 + Z,c y = 2RS. 



For a point-circle at P. 2 (X 2 Y. 2 Z 2 ) 



X./ia + Y a 6/8 + Z,c y = 2KS. 



Hence the radical axis of the pair of circles is 



(X, - X,) aa + (T, - Y,) b/3 + (Zj - Z,) c y = o. 



