where 



62 PROCEEDINGS OF SECTION A. 



If now three lines be taken, viz. : — 



/ ' = 7,a + m,(i + ??,7 — 

 /_" = l^a + m„/3 + «,y = 



then the conic 



Zo' i: Li" 



- + ^- + - =0, c. 



Z ' Z " z '" 



Zo' =^ ^.«o + W'l/S^ + ^*.7o 

 Z o" = ^^ao + ■'"2/5„ + 72,7o 

 Z;"=/3a„+ ?«3^, + «,7e, 



is the locus of points whnse axes of homnlofjy with respect to the 

 triangle formed by /_' L" L '" pass through the point a^^ojo- 



If we express that the conic C, is a circle, we obtain linear equations 

 to determine the co-ordinates of the symmetiian point of the triangle 



L'ri'". 



If the equation of the circle circumscribing the triangle L' L" L'"^ 

 be known, the ratio a^: ji^: y„ may be at once found by comparing 

 similar coefficients in the given equation and the equation C^. 



2. The equation of the circle passing through the three ex-centres 

 Ij, Ij, I3 of the triangle of referei.ce is — 



(a + ;8 + y) («a + i/3 + cy) + fl^y + ^»7a -f ca/3= 0. 

 If this circle be written — 



Po + y o Jo + C ^o ^0+ ^ o _ 



y8-hy y + a a-|-/?~' 



where a^fi^y^ are the co-ordinatts of the pymmedian point of the 

 triangle I.t^Ij, we have, on comparing coefficients — 



whence a^: (3^: yo=b + c — a -. c + a — b : a + b — c 



ABC 



= cot — : cot - : cot —' 



In a similar manner it may be shown that the trilinear ratios 

 of the symmedian points of the triangles IJ J3 ; ITJt : II, I, are, 

 respectively, s : s — c : s — b ; s — c : s : s — a ; s — b : s — a : s, or 



cot — : tan — : Ian — ; tan — : cot — : tan — ; tan — : tan 

 2 2 2 2 2 2 2 



— : cot — . 



2 2 



The four points thus found, together with their isogonal conju- 

 gates witli respect to the triangle ABC, lie on the cubic — 



- W'-y') + 4 (r'-^') + - (a'-/?') =0. 



