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38 11, 00 ART. VIII.-On certain Conic-loci of Isogonal Conjugates. By Everyx G. Iloco, M.A., Christ's College, Christchurch. [Ricart brfore the Philusophical Institute of Canterbury, Ist July, 1908.] 1. The locus of a point P (aby) which moves so that the line joining it to 111 its isogonal conjugate P' passes through a fixed point (wobey) is a • the cubic B (B2 – 7) + B. (72 – a) + (a* -- B“) = 0 1ť, however, the point (Boy) lie on either the internal or external bisector of an angle of the triangle of rcfcrenco, tho cubic becomes a conic and a straight line; and the object of the present paper is to investigate certain properties which this family of conics possesses. For the sake of brevity the fixed point (aya) through which the fiue joining any point to its isogonal conjugato passes will be called the centrum of the conic. The co-ordinates of the centrum are comprised in the system (aq 11) if we limit ourselves merely to the internal and external bisectors of the angle A of the triangle of reference ABO. The following four types of couic exist :- Centrum (0,11) re? + By - (B+ y) = 0 I Centrum (ao --11) - By + aner (B - y) = 0 11 Centrum (4,1-1) Fyx B -- y) = 0 III Centrum (an -1-1) ? + By to ava (+ y) = 0 IV 2. These couics possess the following properties: they all pass through the vertices B and C of the triangle of reference; those oi classes I and IV pass through the ex-centres I., and I, : those of classes II and III pass through the in-ceutre 1 and the ex-centre I. The tangents to the conics at I and I, or at I, and 1, pass through the centrum; the tangents to the conics at B and C meet at the isogonal conjugate of the centrum. llence, 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 roference is parallel to either the internal or external hisector of the angle 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 Pand Q will be at infinity in directions perpendicular 10 the Sinson lines of those points—that is to say, the asymptotic angle of the conic is oqual to the angle lietween the perpendiculars froin the centrum on the Simson lines of P and Q. If the position of the chord of intersection of the conic and circle ABO is deterinined, the position of the asymptotes, and therefore of the By