38 Transactions. 



Art. VIII. — On certain Conic-loci of Isogonal Conjugates. 

 By Evelyn G. Hogg, M.A., Christ's College, Christchurch. 



[Read bpfore the Philosophical Institute of Canterburij, 1st July, 1908.] 



1. The locus of a point P (a^Sy) which moves so that the line joining it to 

 its isogonal conjugate P' ( - 7^ - ) passes through a fixed point (ao^Soyo) 



\ap r/ 



"0 Po 70 



If, however, the point (ao^Soyo) lie on either the internal or external 

 bisector of an angle of the triangle of reference, the cubic becomes a conic 

 and a straight line ; and the object of the present paper is to investigate 

 certain properties which this family of conies possesses. For the sake of 

 brevity the fixed point (ao,8oyo) through which the line joining any point 

 to its isogonal conjugate passes will be called the centrum of the conic. 



The co-ordinates of the centrum are comprised in the system 

 (o-o + 1 + 1) if we limit ourselves merely to the internal and external 

 bisectors of the angle A of the triangle of reference ABC. The following 

 four types of conic exist : — 



Centrum [a^ 11) a^ + /3y — aoa (^ + y) = ... I 



Centrum (ao —11) a^ — /3y + aoa (/3 — y) = ... II 



Centrum (ao 1 — 1) a' — (3y — aga {(3 — y) ■= ... Ill 



Centrum (ao — 1 — 1) a''^ + /3y + aoa (/S + y) = ... IV 



2. These conies possess the following properties : they all pass 

 through the vertices B and C of tlie triangle of reference ; those of 

 classes I and IV pass through the ex-centres L and I^ ; those of 

 classes II and III pass through the in-centre I and the ex-centre Ij. The 

 tangents to the conies at I and Ii or at L and I3 pass through the 

 centrum ; the tangents to the conies at B and C meet at the isogonal 

 conjugate of the centrum. Hence, when the position of the centrum has 

 been assigned, the centre of the conic can be constructed geometrically. 



Furthermore, the chord of intersection of any conic of this family 

 with the circumcircle of the triangle of reference is parallel to either the 

 internal or external bisector of the angle A of that triangle. Suppose 

 any conic to cut the circle ABC in the points P and Q : then, since the 

 isogonal conjugate of any point on that circle lies at infinity in a direction 

 perpendicular to the Simson line of the point, the isogonal conjugates of 

 P and Q will be at infinity in directions perpendicular to the Simson lines 

 of those points — that is to say, the asymptotic angle of the conic is equal 

 to the angle between the perpendiculars from the centrum on the Simson 

 lines of P and Q. 



If the position of the chord of intersection of the conic and circle 

 ABC is determined, the position of the asymptotes, and therefore of the 



