o2 UKITERSITY OF VIRGINIA PL'BLICATIONS 



(16), (17) there result the equations to these cii'cles in areal coordinates. 

 Thus for example the equation to the second circle in (16) is 



= :^(c- + b- — a- — 4 A cot B PC) x — %xyc\ (18) 



cyclic interchanges give the other two. 



The coefficient of x in (18) is known to be tlie power of A with re- 

 spect to tliis circle. Eepresenting this power by /;,, the area of A'B'C 

 by 8. 



p^ = ^{c- + b- — a- — iAeot B PC), 

 = i{c- -\-b- — a- ±: 4 A cot A'), 

 _ (c-4- b- — a^)8 ± {(■'-+ //- — «'-) A 

 ~ 2S ' 



sin (A' ± A) 

 = 2A . (19) 



sm A smA ^ 



The upper sign taken when Q is inside ABC, the lower when outside. 

 Cyclic changes of letters give po, Ps the powers of B, C with respect to 

 circles C P A, A P B respectively. 



The radical axes of these three circles are 



■-*• Pi = y P« = z Ih, (= 5 a- i/ C-), (20) 



and they meet in the point P. Hence the areal coordinates x, y, z of 

 the critical point P are Icnowu, and are 



sin A' sin A sin B' sin B sin C" sin C 

 sin(4'±.i) ■ sin(S'±5y ' sin(G' ± C) ' ^^^^^ 



the upper signs placing P when Q is inside ABC or a, ^, y are positive, 

 the lower sign when Q is outside A B C or when one of a, p, y is negative. 

 Dividing (20) tlrrough by s, each ratio of (20) is equal to each of 

 (11), and therefore 



^•i ^ :^_ = _':^ = J_ (32) 



a]\ I3p„ yp„ S ' 



whence 



s- = a-p, + p-po + y-ps. (23 ) 



= 1(6= _(_ c^ _a=)a'- + i (a- + C- — &=)&'= + i (a- + &= — c^)c'-± 8 8 A, 



the positive sign giving the value of s when Q is inside, the negative when 

 outside ABC. In the above expression for s- the numbers a, b, c and 

 a', b', c' are respectively interchangeable. We may observe that the ex- 

 l^ression 



\{r- + c"- — a-')x"- + 4(fr + c^-—b"-)y-' + H"' + V- — c"-)z% (24) 



