Hogg. — On certain Conic-loci of Tsogonal Conjugates. 39 



axes, may be found, since the centre is known, vvhile the eccentricity of 

 the conic mav be deduced from the relation 



e2- 



2E 



jj + R 



where R is the radius of the cu^cumcircle, and j; is the length of the 

 perpendicular from the cii'cumcentre on the chord of intersection. 



3. We now proceed to deal with certain particular conies of this 

 family. The line f3 — y — o will meet the line at infinity in the point 

 ib + c, — a, — a) ; with this point as centrum we have the conic 



a(a^ + /Sy) + (6 + c)a(/3 + y) =0 A 



which may be written 



a /3y + f3ya + Ca/3 + a {aa + b(3 -\- Cy) = 0. 



It is the cu'cle on IJ^ as diameter. Hence the theorem — 



" Any line parallel to the internal bisector of the angle A of the 

 triangle ABC meets the circle described on I.^Ig as diameter in two points 

 which are isogonal conjugates with respect to that triangle." 



The external bisector /3 + y = o meets the line at infinity in the point 

 ih — c, —ft, ft). This point being taken as centrum, we have the conic 



a{a' - fSy) + {b -C)a{(i -y) ^0 B 



which is the circle on the line IIj as diameter. Hence the theorem — 



" Any line parallel to the external bisector of the angle A of the 

 triangle ABC meets the circle described on IIi as diameter in points 

 which are isogonal conjugates with respect to that triangle." 



4. The line (S — y — o meets the circle ABC in the point ( — ft, b -\- c, 

 Z> + c) : with this centrum we have the conic 



(6 +C) (a-^+ ^y) + aa(/3 + y) = C. 



It meets the circle ABC along the line 



{b + C) (fta - CyS + by) + ft" (/3 + y) = 0, 



which may be written 



{b + c) (fta + 6/5 + cy) - [(/; + cf- ft-^] (/3 + y) = 0. 



This is satisfied by the co-ordinates of the centre of the circle ABC. 

 Hence the chord of intersection is the diameter parallel to the external 

 bisector of the angle A. Therefore, since the Simson lines of the ex- 

 tremities of a diameter of a circle are at right angles to each other, we see 

 that the conic is a rectangular hyperbola. The tangents to the conic at 

 B and C are parallel to the internal bisector of A, hence the centre of the 

 conic is the middle point of BC. 



The li)ie /3 + y = o will meet the circle KQG in the point 



(ft, -b + c,b - c); 



with this centrum we have conic 



ih _ ,) (a2 _ ^^) + aa (^ - y) = D. 



Its chord of intersection with the cii'cle ABC is the diameter parallel 

 to the internal bisector of the angle A, and its centre is at the middle 

 point of BC. 



