676 Transactions. 



9. Let D'E'F', D"E"F" be two centroidal triangles, and let the two 

 corresponding (BC) sides— i.e., those which touch S' = o — meet in the 

 point D (a^^oyo)- 



The equations of E'F', E"F" will be 



X'^cy + k'aa + h/3 = 0] 



X"^Cy + A"aa + b^ ^ o'i (i) 



when A', A" are the roots of the equation 



A^cyo + Aaoo + '^Ao = 0. 

 The sides F'D', F"J3" will be 



A'Vta + \'b/3 + cy = 0) 



A"2oa + A"6/3 + Cy = ) (ii), 



and the sides D'E', D"E" will be 



X'^b/S + X'cy + aa = [ 



A"26^ + A"cy + aa= o\ (iii). 



Let F'D', F"D" meet in E, and D'E', D"E" m F. The co-ordinates of 

 E are given by 



aa : A/3 : cy = 1 : - (A' + A") : A'A", 



butA' + A"=: -^^"-^A'A"^^", 



and therefore 



,. c a b ^ 



"■ ■ P ■ Y = -Yo : ^^o '■ jPo- 



In the same way it may be shown that the co-ordinates of the point F 

 are('^o,p'o,~ao). . . 



The equations of DE, EF, FD are respectively 



aa P„ H- i^ Q^ + cy R^ = 



aa R„ -t- 6^ P„ + Cy Q„ = 



aa Q,_ + b(i R^ + cy P^ = 

 where ^,= a'-a,' - bc(i^y^, Qo^^W - cay„a„ RoEEC^o' - «K^o- 

 Hence if the point (a^fi^y^) lie on any of the conies S^, Sj, S3, the sides 

 of the triangle DEF will pass through the vertices of the triangle ABC. 



We also see that if (a^^^yj lie on the curve /(a/3y) = 0, then E and F 

 will lie on the curves/ h, ^, ^J) = -.ind fC'^, '^ , ^) = 0. 



If the point D (a^,/?^y.) lie on the Steiner ellipse — + A + - = ^'- 1^^®" 



^ ^'"^ ^ (la 0/3 cy 



the points E, F will also lie on this ellipse. In this case we have 



Po : Qo : Ro = ''«o : ^^^o : cy^, 

 and_ it may be easily shown that the sides of the triangle DEF touch this 

 conic 



Vaa + Vb/S + Vcy = 0. 



If D lie on the conic Vaa + VFft + Vcy = 0, so also will the points 

 E and F, and the sides of the triangle DEF will envelop the conic 



VtJ + \/V + \/w= «' 

 where U = — «a -f b/3 + cy 

 V = aa - bfS + Cy 



W = aa + b^ - Cy. 



