334 Transactions. 



then the Simson line of the point P is perpendicular to the 

 line la + m(3 + ny = o, and its equation may be written 



alb c\ blc a\ c/a b' 



~L\m~n) m\n 7/ „ n\l nij 



-jur a+ sir P + -^r~ y = 



dl dm dn 



where 



Q, = P-\-m 1 -\-n 2 — Qmn cos A — 2nl cos B — 2lm cos C. 



5. The isogonal transformation of a tangent to the circle 

 ABC is a parabola circumscribed to that triangle. If the 

 tangent touch at the point P, then the axis of the parabola is 

 perpendicular to the Simson line of the point P. The isogonal 

 transformations of two tangents TP, TQ are two parabolas 

 having their axes inclined at an angle equal to that at which 

 PQ cuts the circle ABC. Parallel tangents transform isogon- 

 ally into two parabolas passing through four concyclic points 

 and having their axes mutually perpendicular. 



6. Let four points A, B, C, D (no three of which are 

 collinear) be taken, and let the triangles formed by omitting in 

 turn each of the points be called Aj, A 2 , A s , A 4 : let also the 

 isogonal conjugates of A, B, C, D with regard to the triangles 

 A 1; A 2 , A 8 , A 4 be called respectively A', B', C, D'. If the tan- 

 gents from A', B', C\ D' touch the circumcircles of Aj, A 2 , A 8J 

 A 4 in PjQi : P a Q a : P 3 Q„ : P 4 Q 4 , then the two parabolas which 

 can be drawn through the four given points may be regarded 

 as the isogonal transformation of any pair of tangents to the 

 corresponding circumcircle. Hence we see that the eight 

 points of contact of the tangents may be arranged in two 

 groups of four points such that the Simson lines of the points 

 of each group are parallel to one another. This result may 

 also be expressed by saying that each of the chords of contact 

 PQ cuts its associated circle at the same angle — viz., the angle 

 at wiiich the axes of the parabolas are inclined to each other. 



7. If the direction of the axis of a parabola circumscribing 

 the triangle ABC be given, the line of which the parabola is 

 the isogonal transformation may be constructed in the follow- 

 ing manner: Draw through A a chord AA' perpendicular to 

 the given direction : let the perpendicular from A' on BC meet 

 the circle ABC in the point P, then the tangent at P to the 

 circle ABC will isogonally transform into a parabola whose 

 axis is perpendicular to the Simson line of P, and therefore 

 parallel to the given direction. 



8. Let a straight line ~L = la + ??t/3 + ny = o be taken, and 

 let p be its distance from the centre of the circle ABC and <f> 



