Hogg. — Isogonal Transformations. 337 



at the circum- and ortho-centres of the triangle of reference 

 to be 



sin A y'a cos A + sin B \J ft cos B + sin C \/y cos C = o 

 This conic has the nine-point circle of the triangle as its 

 auxiliary circle, and its eccentricity is 



Vl — 8 cos A cos B cos C 



If three conies be inscribed in the triangle of reference 

 (supposed acute), the middle points of the perpendiculars from 

 the vertices on the opposite sides being each a focus of one 

 conic, then the major axes of the conies all pass through the 

 centroid of the triangle. 



11. The polar of any point with respect to a rectangular 

 hyperbola self-conjugate with respect to the triangle of refer- 

 ence passes through its isogonal conjugate. Taking the 

 equation of the hyperbola to be la? + w/3 2 + n-f = o, where 

 I -f- m +'w = o, the polar of atiy point P (a'fi'y) is la' a 



+ mft'ft + ny'y = o, which passes through P' ( — , — , — • ) . 



\a ft y / 



Let the polars of P and P' intersect in P", then the 



/U' V W'\ 

 co-ordinates of P" are ( — , — , — , where U = a (ft 2 — y' 2 ), 



V I m n ) 



V = ft (y 2 - a 2 ), W = y (a 2 - ft 2 ). Hence the point P" lies on 



U' V , W 

 a ft y 



a conic which passes through P and P'. Since its equation 

 is independent of /, m, n, we derive the following theorem : 

 Given a fixed triangle and a fixed point, the locus of the 

 intersection of the polars of the given point and its isogonal 

 conjugate with regard to rectangular hyperbolas having a 

 given self-conjugate triangle is a conic passing through the 

 vertices of that triangle, the given point, and its isogonal 

 conjugate. 



The tangent at any point of the rectangular hyperbola 

 la?-\-mft i -\-ny 1 = o passes through its isogonal conjugate. If 

 O be the centre of the hyperbola, and if its asymptotes meet 

 the circle ABC again in the points X, Y, then these points are 

 the isogonal conjugates of the points in which the hyperbola 

 is touched by its asymptotes : hence the diameter XY of 

 the circle ABC will isogonally transform into a rectangular 

 hyperbola whose asymptotes are parallel to those of la? + mft 1 

 + ny 2 = o. 



The equation of XY is easily found to be 



I m n 



- (Cft + by) + j (ay + Ca) + - {ba + aft) = 



