Hogg. — On certam Conic-loci of Isogonal Conjugates. 41 



7. If the centrum be taken at the point (Oil) in which the internal 

 bisector of the angle A meets BC we have the conic 



a' + (3y = o L 



touching AB, AC, at B and C respectively, and passing through L and I3. 



This conic meets the circle ABC in the line 



aa — c/3 — by = 0, 



showing that it passes through the point in which the tangent to the circle 

 x\BC at A meets BC. This chord may also be written in the form 



«a + 6/3 + Cy - (i + C) {(S + y}= 0, 



showing that it is parallel to the external bisector of the angle A. It has 

 real intersections with the circle ABC, hence the conic is a hyperbola. 



The tangents at L and I., meet at the centrum : hence the centre is the 

 point in which the median drawn from A meets the line joining the 

 centrum to the middle point of I2T3. The position of the axes of the conic 

 is therefore given. 



8. If the centrum be taken at the point (0-11) in which the external 

 bisector of the angle A meets BC we obtain the conic 



a2-/3y=0 M 



which intersects the circle ABC along 



aa -\- bfi -\- cy — {b — c) (yS — y) = 0, 



a line which passes through the intersection of BC and the tangent to the 

 circle ABC at A. The intersections of this line with the circle ABC are 

 real if a^>4ibc, in which case the conic is a hyperbola. If a^ = 46c, the 

 conic is a pai'abola whose axis is perpendicular to the Simson line of the 

 point in which the above line touches the circle ABC. If a^<ibc, the 

 conic is an ellipse : to determine the direction of its axes, draw through 

 the pole of this line with respect to the circle a line parallel to it, then the 

 equi-conjugate axes of the ellipse are perpendicular to the Simson lines of 

 the two points in which this line cuts the circle ABC, and the directions 

 of the axes are therefore obtained. 



9. The conies of this family possess the property that the isogonal 

 transformation of the tangent to a conic at any pomt P is a conic circum- 

 scribing the triangle of reference and touching the given conic at the 

 point P', which is the isogonal conjugate of P. 



Hence one of the coumion tangents to the circles described on Ijl.,, 

 Ijlg as diameters will be a circumconic which touches each of these 

 circles at the points isogonally conjugate to the points of contact of the 

 common tangent. 



10. Let any straight line L meet the circles described on the lines 

 I.^Ig, L3I1, IJ.j as diameters m the three pairs of points PiP,2, P3P4, PaP,; 

 respectively ; let chords of the three circles be drawn through PjP., 

 parallel to the internal bisector of A, through P.^Pj parallel to the internal 

 bisector of B, and through P^P,- parallel to the internal bisector of C; 

 and let the extremities or these six chords be respectively P/P.a', P3'P4', 

 Pj'Pfi'. Further, let the line L cut the three circles on IIj, II.,, II3 as 

 diameters in the pairs of points Q1Q.2, Q3Q4, QsQe, and let chords of these 

 circles drawn parallel to the external bisectors of A, B, and C respectively 

 have their extremities at Qi'Q.2', Qs'Q4', Qs'Qe'- Tlien the fifteen points 

 A, B, C, the six points P', and the six points Q' all lie on the conic 

 which is the isogonal transformation of the line L. 



