494: Transactions. 



The equations of the lines joining P to the isogonal conjugate of F 

 and P' to tiie isogonal conjugate of ¥' are respectively 



IH'ax + m^m'by + nhi'cz = o 

 ll'^ax + mm'^by + nn''^cz = o. 



These lines, which are parallel to the axes of S and S' respectively, 

 intersect on the Steiner ellipse of the triangle ABC in the point 



-— ,, : ;, 7 ), which is the centre of perspective of the triangle ABC 



all bmm cnn'/ 



and the "contact" triangle of the inscribed parabola which has its focus 



at the trilinear pole ( ,— , ,, — - ) of the chord FP'. 



\U vwi nn / 



The equation of the trilinear polar of the isogonal conjugate of F' is 



^r H — -.-{ — 7 = 0; from its form it is seen that it touches S and also the 

 V m n' 



maximum inscribed ellipse, 'V ax + \^h^j + \cz = 0, of the triangle ABC. 



The corresponding fourth common tangent of this ellipse and the para- 



(XX uTJ CZ 



bola S' is -p + ~ H = ; the trilinear ratios of the point of inter- 



i m n 



section of these lines are (— , — j— , - — ), which satisfy the equation of 



the line at infinity. Hence these common tangents are parallel ; they 

 touch the ellipse at the extremities of the dianieter whose equation is 

 Wax + mm'by + nn'cz = 0; i.e., the diameter of the Steiner ellipse whose 

 extremities are the centres of perspective P, P'. These common tangents 

 to the maximum inscribed ellipse and the parabolas touch S, S' at the 



fll'^ mm"' nn'^\ flH' mhn' nhi'\ . , , , ,. . . 



pomts I — , — -, — , I, I — , — ; — , I respectively, and the Ime join- 



(TCC 01.1 CZ 



ins these points — viz., -jj-, -\ , -\ , = — is a tangent to the maxi- 



° ^ W mm nn ^ 



mum inscribed ellipse. 



5. The equation of the lines joining A to the points in which S and 

 S' touch BG is 



(mby — 7icz) {m'hy — n'cs) = ; 



i.e, mm'b-y'^ + nn'c^z^ — {inn^ + m'li) bcyz = 0, 



or mm'b'jf- -\- nn'c-z'^ — 2ll'bcyz = 0. 



Hence the six points of contact of S and S' with the sides of the triangle 

 ABC lie on the conic 



Si^ZZ'rt^x^ + mm'b'^y^ + nn'cV — lll'bcyz — 2mm'cazx — 2nn'abxy = 0. 



If in Si we substitute for {x, y, z) the trilinear ratios of the isogonal 



/•I r • T-i T-r • (I m n\ (V m' n'\ 

 conjugates of the toci b, !:• — viz., I -, -j-, -1, I -, — , —I — we obtain the 



expressions IH' -f m^m' + 7ihi' — 2Zw« (/' + m' -f n'), 



IV^ + mm'* + nn'^ - ll'm'n' {I + m + n), 



both of which vanish ; hence the asymptotes of Sj are parallel to rhe 

 axes of S and S'. 



