IIoGQ. — On Centroidal Triangles. G75 



Hence the triangle ViVgV:! is in perspective with the triangle ABC, the 

 centre of perspective being the isogonal conjugate of the point 



[{he + a^ cos A), {ca -f- b^ cos 13), {ah + c^ cos C)] , 



which is the centre of the Brocard circle. 



If p^, 2hy Ih be the lengths of the perpendiculars from V^.V.^.V;, respec- 

 tively on the opposite sides of the triangle ViV.jV^, then 



^•^1^1 = VA = V^^ 

 the common value of these products being 



\ [2 {a') ^ D^- (1 + cos A cos B cos C)l . 



If a^, b^, Cj be the lengths of the sides of the triangle ViV^V.,, then 



Hence the sides of the triangle formed by the directrices of the three 

 parabolas S', S", S'" are proportional to the medians of the triangle ABC. 



8. Writing X for - we have as the equations of the sides of a centroidal 

 triangle 



X^ Cy + X (la + bfS = 

 X^ Cla + X b/3 + Cy = 

 X^ h/3 + A Cy + rta = 0. 



Let the condition now be determined that a triangle (AoBiC.,), the pai'a- 

 meter of whose sides is X,, maybe inscribed in a triangle (A^B^Cj) whose 

 sides have the parameter A^. 



Solving for the equation of the sides CgAg, A3B2 ^or the co-ordinates of 

 the vertex Aj we have 



aa : b/3 : Cy = : — 1 : A.^. 



Hence if Ao, lie on B^C^ we have the condition 



AiU2= 1. 



The same condition holds that Bo and Co lie on C^A^ and A^B^ respec- 

 tively. 



We may now find the locus of the intersection of the corresponding 

 sides BjCi, BA 



A^ aa -f hft + Aj- Cy = 



Ao aa -{- b/3 -\- X./ Cy = 0, 

 and therefore 



aa : b/3 : Cy = — (A^ + Ao) : X^X^ : 1. 



Eliminating A^ and Aj between the above and the equation A^^Aj = 1, we 

 see that the intersection of corresponding sides (BC) lies on the cubic 



b'/3' + cV + abca/3y = 0. 



Similarly the intersections of corresponding sides (CA) and (AB) lie on 

 the cubics 



cy + ftV" + abca/3y = o 



af'a:' -f ¥p^ -f- abca/3y — 

 respectively. 



