Hogg. — On Orthogonal Circles. 585 



Suppose the centre of the orthogonal circle to lie on the line 



pa + qB -f r-y = o at the point (-,-,- ), where A + u + v = o. The 



\p q r J 



equation of the circle will be 



^X + > xb Y + vC Z = o, 

 p q r 



showing that the circle passes through the pair of inverse points deter- 

 mined by the intersections of the circles 



aX bY cZ 



p ~ q ~ r " 



Hence the theorem "All circles having their centres on a given line 

 and cutting a given circle orthogonally pass through two fixed points 

 which are inverse with respect to the given circle." 



Let A^C 1 be a triangle inscribed in the circle ABC and circum- 

 scribed to the Brocard ellipse of the triangle ABC. 



If X + fx + v = o, the trilinear ratios of A 1 , B 1 , C 1 may be taken as 



r-i -> -)> (-. -i v)> (-' v -) respectively, for each of these ratios satis- 

 A fj. v) \fji v A/ \v A fxj 



fies the equation of the circle ABC, and the line joining B 1 ^ is 



~ + £- + % = o, the envelope of which is V - + V r + ▼ - = o. 

 aX Ofx cv a b c 



Let three circles be described cutting the circle ABC orthogonally, 

 each circle passing through two vertices of the triangle A^C 1 . The 

 centre of the circle through B^ 1 is the pole of B 1 ^ with respect to the 

 circle ABC, hence the trilinear ratios of the centre (a 6 y ) are 



H-H+'Wx-HM^H)] 



and the tripolar equation of the circle is 



( _j + MW + (i-MW + (M-i)«-.; 



\ A jx vf \A fx v / \A fx v J 



i.e., -S^-Sa-f-Sg = o, 



A fX v 



where Si = - a 2 X + b l Y + o 2 Z. 



Hence the envelope of the system of orthogonal circles is 



Vs; + \/sT 3 + VS 3 = o. 



It may be remarked that Sj = o is the equation of the locus of points 

 whose pedal triangles are right-angled, the vertex of the right angle lying 

 on BC. 



The radical centre of the three orthogonal circles is the svmmedian 



point of the triangle ABC, for if L, =— + ^- + --, L 2 = -- -f f- -|- X 

 r ° aX. b/x cv a/x bv ck 



L, = — + ~ + — , then the radical axes of the circles in pairs are 

 av bX Cfx 



L 2 — L 3 = o, L 3 — Lj = o, L 2 — L 2 = o, 



and it is at once seen that the equations of these lines are satisfied by the 

 co-ordinates (a, b, c) of the symmedian point. 



