Hogg. — Isogonal Transformations. 333 



Art. XXXI.— On Isogonal Transformations : Part II. 

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



[Read before the Philosophical Institute of Canterbury, 4th December, 



1907.} 



1. If from a point P perpendiculars PD, PE, PF be drawn to 

 the sides BC, CA, AB respectively of the triangle of reference 

 ABC, it is easily shown that the perpendiculars from A, B, 

 and C on EF, FD, and DE respectively are concurrent in a 

 point P', and that the points P and P' are isogonal conju- 

 gates. 



If now the point P be supposed to move on to the 

 circle ABC, the point P' will move to infinity, and the pedal 

 triangle DEF will become the Simson line of the point P. 

 Hence we derive the important theorem — " The isogonal con- 

 jugate of a point on the circumcircle of the triangle of 

 reference lies at infinity in a direction perpendicular to the 

 Simson line of the given point." 



2. In this paper use will also be made of the following 

 theorem : " The Simson lines of the extremities of a chord of 

 a circle intersect at an angle equal to that at which the chord 

 cuts the circle." This may be easily proved from the con- 

 sideration that if the perpendicular drawn from any point P 

 on the circle ABC to BC meets that circle again in the 

 point A', then A A' is parallel to the Simson line of P. 



3. It has been shown in section 4 of Part I of this paper 

 that the asymptotic angle of the circumconic which is the 

 isogonal transformation of a chord of the circumcircle of the 

 triangle of reference is equal to the angle at which that chord 

 cuts the circle. Combining this with sections 1 and 2 of this 

 paper, we see that the asymptotes of the conic which is the 

 isogonal transformation of a chord PQ of the circle ABC are 

 perpendicular to the Simson lines of the points P and Q. 



In general, if S' be the isogonal transformation of a curve 



S, and if S cut the circle ABC in the points P, Q, E , 



then the directions of the asymptotes of S' are perpendicular 

 to the Simson lines of the points P, Q, B 



4. If the position of a point P be determined by the inter- 

 section of the circle ABC and the conic whose equation is 



Ifiy + niya -f Uaft = 0, 



