490 Transactions. 



Art. L. — On the Inscribed Parabola. 



By E. G. Hogg, M.A., F.R.A.S., Christ's College, Christchurch. 



[flead before the Philosophical Institute of Canterbury, 1st September, 1915.] 



1. If I -\- m -\- n = 0, the conic inscribed in the triangle of reference 

 ABC, whose equation is 



S ^ Vlax + Vmby + V^tcz — o, 



is a parabola whose focus F has trilinear ratios (y, — , -). 

 ^ ♦ \/ m nj 



It may be remarked that if a line cutting BC, CA, AB in D, E, F 

 respectively move so that the ratio DE : DF — p: q, the envelope of the 



line is the parabola V{p — q) ax + V — pb^ + Vqcz = o ; 

 i.e., I : m : n = p — q : — p : q. 



The equation of the directrix of the parabola is I cos \x + m cos Bt/ 

 + 71 cos Gz = 0. 



The trilinear polar of F and the directrix of S meet on the conic which 

 is the isogonal transformation of Euler's line [i.e., the line passing through 

 the orthocentre, centroid and circumcentre). 



Let S touch BC, CA, AB in A', B', C respectively : the equations of 

 B'C, CA', A'B' are respectively 



L = — lax + mby + ncz = o 

 M = lax — mby + ncz = o 

 N = lax + mby — ncz = o. 



The lines L, M, N pass through the fixed points ( > i:> ~)' ( ~' ~ a' ")' 



{-, J, ) respectively, and the triangles ABC, A'B'C are in perspective^ 



the trilinear ratios of P — the centre of perspective — being ( — , — 7, — |. 



The locus of P is the Steiner ellipse of the triangle ABC {i.e., the 

 circumscribed ellipse whose centre is at the centroid, or maximum 

 circumscribed ellipse). The line FP passes through the fixed point, 



t~7z:5 ^» rT"5 ST' -T~5 ttt ) which is a point of intersection of 

 a (0^ — c^) (c- — a^) c {a^ — b-)J ^ 



the circumcircle and Steiner ellipse of the triangle ABC. 



The equation of the maximum inscribed ellipse of the triangle A'B'C 

 is Zv L + mv M + 7iV N = ; hence P is also situated on this conic. 



The bisector of the diagonals of the quadrilateral formed by the lines 

 L, M, N and the trilinear polar of P — viz., lax + mby + ncz = — is the 

 line Pax + m^by + n'^cz = 0, which touches its envelope — the Steiner 

 ellipse — at the point P. 



