IIoOG. — On Centroidnl Trianr/hs. 673 



Let lines drawn throuf,'li A, B, C parallel respectively to L'M'N' meet 

 the opposite sides of ABC in DM^VF'. The equation of the line-pair 

 AX', AD' will be 



{php - qcy){qhf3 - pCy) = 

 pq (^2/^2 ^ c2y2) - (p^ + q^) hefty = 0. 



Hence the six points X'Y'Z', D'E'F' lie on the conic 



— (p2 -|- q^){bc-[iy + caya + aha ft) = o. 



We now proceed to show tliat this is the Steiner ellipse of the 

 triangle X'Y'Z' — i.e., the locus of points whose polars with respect 



to the triangle X'Y'Z' pass through the point G. If -, r, - be substituted 



for a, ft, y in Lj, Mj, Nj, these quantities have the common value 

 2:>~ — 2^q + q^ 'y hence the equation of the Steiner ellipse of the triangle 

 X'Y'Z' may be written 



1,1,1 

 Li + M, + Ni = ^- 

 This on multiplying out and dividing by the common factor 2^~ — pq 

 + q- reduces to S^. 



Let lines through ABC parallel to Lj^MjNj respectively meet the 

 opposite sides of that triangle in D^E^F^. Then the six points A'B'C, 

 DjEjFj lie on the conic 



So' = 2^1 («'«' + ^'/Q' + c'y') 

 + {v"^ + q^}{^(^f^y + ca-ya + ahaft) = o, 

 and this conic is at once shown to be the Steiner ellipse of the triangle 

 A'B'C. 



The envelope of the Steiner ellipses of centroidal triangles, as the 

 ratio 2^ '• q varies, is 



{aa + hft + cy)'^ {Vau + Vbft + Vcy) = 0. 



6. The circum-circle of the triangle AY'^'Z' will for all values of the ratio 

 2) : q pass through a fixed point. The equation of the circle in question is 

 p [{a^ - h^) fty + caaft - bcy^] 



— q [{c^ — a') fty — ahya + hcft"^] — o. 



Hence this circle passes through the intersection of two fixed circles, 

 which may be written 



«S — hyh = o 



rtS — cftLi = 

 where S = afty + bya + ca(3 and L =^ «a + hft + cy. 



The former of these circles touches AC at A and passes through B ; 

 the latter circle touches AB at A and passes through C. The radical 

 axis of these circles is eft — by = o, and this line meets the circles again 



in the point H' (- ^ — , h, c\. 



Hence since Y'Z' envelops a parabola S' which touches AY' and 

 AZ', and the circum-circle of AY'Z' always passes through a fixed point 

 H', that point must be the focus of S' = o. 



Similarly it may be shown that the foci of the parabolas S', S" are at 



^1 ■ i. TT// / 2crt cos B \ rrn, / 7 2a6 COS C\ ,• 1 



the pomts H" ( a, r , cj , H" I a, h, \ respectively. 



