Hogg. — On the Inscribed Parabola. 497 



8. If L = 0, M = 0, N = be any three lines, the equation of the 

 parabola inscribed in the triangle formed by them is 



^/flh + s/^rmM + \/hnl^ = o, 

 where I -\- m -\- n — o, and the equation of the line at infinity is 



/L + gU + /iN = 0. 

 The equations which determine the focus are 



ZL mil _ nN 

 fn^ ^ 'ga^ ~ Mis' 

 and the equation of the directrix is 



flh (-/'Oi + c^^flj + h^Qs) + gmM {fn, - g^^ + h^a,) + /iwN 



(/-Oi + (f^2 - h'n,). 



The trilinear ratios of the centre of perspective of the triangle LMN 

 and the triangle formed by joining the points of contact of the parabola 

 with the sides of that triangle are given by the equations 



flLi = gmM = /wiN, 



and these ratios satisfy the equation of the Steiner ellipse of the triangle 

 LMN— viz., 



/L + gM + /iN ~ ^- 



The trilinear ratios of the isogonal conjugate of the focus is found from 

 the equations 



fh_g_M _hN 



I m n ' 

 and the equation of the axis of the parabola is 



The equations of the sides of the " contact " triangle are 



Li = - flh + gmU + hnl^ = o 



Ml = flh - gmM. + hn^ = o 



Ni = fllj + gmM - huB — o, 

 and the equation of the parabola may be written 



1 1 1- 



l; + m; + n; = ^- 



