Hogg. — On Isogonal Transformations. 309 



7. Any triangle circumscribing the conic 



S = A./3y + /xya + va/3 = 



will invert into three circumconics having the line 



L = Aa+/x/3+vy = 

 as a common tangent. 



A family of n parabolas circumscribing the triangle of 

 reference will invert into an 7i-sided polygon in which the 

 circle x\BC is inscribed. 



The pencil of lines represented by the equation 

 l^a-\-mi(3-j-niy + K {l^a + nii/S + v^y) = 



where k varies, will invert into a family of conies passinig 

 througli the four points of intersection of the conies 



llfty + Vl•^ya-\-n■^a|B = 



hPy-\-t''2yo--{-iho-P = 

 In particular a system of parallel lines will invert into 

 a family of conies passing through four concyclic points. 



Hence, as there will always be two lines, whether of the 

 pencil or of the parallel system, which are equidistant from 

 the centre of the circle ABC (excluding those lines of either 

 system which are diameters of this circle), we see that all 

 conies passing through four given points may be arranged in 

 pairs of similar conies. 



8. Two tangents drawn from a point P to the circle x\BC 

 will invert into two parabolas passing through ABC and P'— 

 the inverse of P with respect to the triangle ABC. 



Hence if four points, ABCD, be given, and if A' B' C D' be 

 respectively the inverses of those points with respect to the 

 triangle formed by joining the remaining three points, we see 

 that the two parabolas which may be drawn through four 

 given points can be regarded as originating by inversion of 

 the pair of tangents from the four points A' B' C D' to the 

 circles BCD, CDA, DAB, ABC respectively. 



Now, if one of the points, say D', fall within the circle 

 ABC, the tangents from it to that circle are imaginary, and 

 consequently the two parabolas through ABCD are imaginary : 

 therefore the remaining points A' B' C must lie within the 

 respective circles BCD, CDA, DAB. 



We may state this result as follows : If any four points 

 be taken on a parabola, the inverse of any one of the points 

 with respect to the triangle formed by joining the remaining 

 three points lies without the circumcircle of that triangle. 



