Hogg. — On Centroidal Triangles. 671 



If the sides of the triangles X'Y'Z', A'B'C be respective!}^ rr', y' , z' 

 and a' , h' , c' 



Ma-)='Hj^-Ma') 



{p - q)' 



If 2h> lh> V-' ^® ^^® perpendiculars from G on the sides of a centroidal 

 triangle, and^', if, p'" the perpendiculars on those sides from A, B, C, 

 then 



;_^ _ Pa _ i^s 



p' p" p'" 



The equation of the circle circumscribing any centroidal triangle is 



(A -1)(A^' - 1) abc {a/3y + bya + Caft) 

 + A {an + b/3 + cy) [aa (a-A + b'X^ + c'^) + b^ {a^ + &-A + c^A-) + Cy (a-A'^ 



+ Z'--^ + C'X)] := 0, 



and the radical axis of this circle and of the circle ABC envelops, as A 

 varies, the conic 



{a' - 46V^) aV- + {b' - 4:chi:') 6-/5-' + (c* - Aa'^b-) cy 

 - 2 (2a^ + b'c^) bc^y - 2 (26^ + c'^a'^) caya - 2 (2c^ + d'b'^) aba/S = o. 



The locus of the symmedian point of the triangle xAY'Z' is the curve 

 2bc (cf3' + by') - c(c^ - a") fiy + b {a^ - b'') (if = abcdfty. 



3. The equations of the lines B'C, C'A', A'B' are respectively 



L' = pqaa + q'^b/3 + p-cy = o 

 M' = _p^aa + 2^q b/S + q~cy = o 

 N' = q^aa + p~b^ + P^cy = o, 



while those of Y'Z', Z'X', X'Y' are respectively 



Li = — ^jg aa + g-ftyS + p^cy = o 

 Ml = 2faa — pqb/3 -{- q^Cy = o 

 Nj = q^aa + ^~6y8 — pqcy = o. 



Hence as the ratio jj : q varies, the lines L', Li; M', Mj; N', Nj envelop 

 respectively the parabolas 



S' = aV - 46c^y = 

 S" = 62;82 _ 4g^^^ ^ 



S'" = cy - 4ft6a^ = 0. 

 The points of contact of L' and Li with S' are respectively 





hence the sides of any centroidal triangle are divided internally and 

 externally in the same ratio at the points in which they touch their 

 enveloping parabolas. 



