Hogg. — On Orthogonal Circles. 



589 



Let Uj be the result of substituting in (i) X^Zi for X, Y, Z 

 respectively. Let a circle concentric with U = O pass through the point 

 whose tripolar co-ordinates are (X 1 Y 1 Z,), and let r' be the radius of this 

 circle, then 



U x = aa.X, + 6&Y, + cy Z x - 2RS - 2 a r 2 



O—aa^ + b^Y, + cy^ - 2RS - 2Ar' 2 , 



and therefore Ui = 2 A (r' 2 — r 2 ) = 2 Af 2 , where t is the length of the 

 tangent from (XjYxZj) to the circle U = O. 



Let t lt t 2 , t 3 be the tangents to the circle aa X + bft Y + cy Z = o 

 from A, B, C respectively, then 



2a^ 2 = bp o c 2 +c 7o b 2 



+ cy a 2 



2 A * 2 2 



aa c 



Hence 



2Ai 3 2 = aa 6 2 + 6/? a 2 



2a =aa +6)8° +cy 



^ 2 o c 2 6 2 



£> 2 c 2 o a 2 



o, 



t 2 6 2 a 2 o 

 1111 



which, on expansion, gives the following relation connecting the lengths 

 of the tangents from the vertices of a triangle to any circle cutting 

 orthogonally the circumcircle of the triangle, viz., — 



t? sin 2A + t? sin 2B + t 3 a sin 2C = 4 a . 



If t', t", t'" be the lengths of the tangents to an orthogonal circle 

 from the middle points of the sides of the triangle ABC, then 



2Ar 2 = aa m 1 2 + | 2 (5/3 + cy ) 

 •2 A t" 2 = bfoll 2 + -r (c 7o + aa ) 



2At'"* = cy m a * + I (aa + bft), 



where m u m 2 , m 3 are the medians of the triangle ABC. Hence 



2 a (t' 2 + t m + t"'*) = aa Q \m? + — f-j + 6 & ( w 2 2 + — ~— j 



2 , « 8 + & 2 



= aa. 



3 (6 2 + c 2 ) 



+ cy (ni? + 



7 „ 3 (c 2 + a 9 ) - b 2 3 (a 2 + b 2 ) - c 2 



; ) 



+ *& 



+ Cy 



4 - -r-o 4 - -,o 4 



also 2 a (V + C- + f 3 2 ) = aa Q (b 2 + c 2 ) + b/3 (c 2 + a 2 ) + cy (a 3 + 6 2 ). 



Hence 



2 a 1 2 it 2 ) - 2 (t' 2 ) \ = j (aa Q + b£ + c 7o ) 2 (a 2 ) ; 



i.e., 



2(t 1 *)-S(t'*)=±3 l ( a *). 



