' Hogg. — On the Inscribed Parabola. 491 



The triangle A'B'C is self-conjugate with respect to the Steiner 

 ellipse, whose equation may be written IJJ + r)iW' + nW — o. 



The polar of the focus F Ij, -, - j with respect to the triangle A'B'C 



has for its equation 



L M N _ 



fe2 + c^ - a- "^ c^- + a;' - b^'^ a^ + 6'- - c^ ~ ° ' 

 i.e., tan AL + tan BM + tan CN = o, 



which may be written in the form 



m{(tan B + tan C — tan A) ax — (tan C + tan A — tan B) by} 

 + n {(tan B + tan C — tan A) ax — (tan A + tan B — tan C) cz) = o, 

 showing that the polar passes through a fixe^ point. 



The sides of the medial triangle of the triangle A'B'C envelop 

 parabolas as F moves on the circle ABC. The equation of the side of 

 the medial triangle parallel to B'C is — /'^L + m^M + w^N = o, which 

 reduces to 



(m^ + m7i + n^) ax + 7n'^hy -\- ii^cz — o, 



and the envelope of this line is the parabola 



a^x^ — 4 (ax -\- by) {ax + cz) . 



If S be expanded and — I {m + n), — m {n + I), — n {I + m) be sub- 

 stituted for l^, m^, n^ respectively, we have 



S = mn {by + cz)^ + nl {cz + ax)''^ + Im {ax + by)^ = o, 



showing that the triangle whose medial triangle is the triangle ABC is 

 self-conjugate with respect to S. 



2. The line at infinity expressed in terms of L, M, N is Ph + w^M 

 -(- li'^N = ; if {x'y'z') be the trilinear ratios of the centroid G' of the 

 triangle A'B'C, then the polar of G' with respect to the triangle — viz., 

 L M N 

 =p +'^» "^ N* ~ ^ — ^^ identical with Z'^L + m^M + «'^N = o. Hence Z^L' 



= m-M' = nm' = K, and therefore M' + N' = llax' = k ( — + ^ ), whence 



ax' _ by' _ . cz' 

 I {w? + v?) ~ m {n^ + P) ~ n {P + m^) 



The isogonal conjugate of F — viz., the point ( -, y , - ) — lies at infinity in 



a direction perpendicular to the pedal line of F, and therefore parallel to 

 the axis of S. The line joining G' to the isogonal conjugate of F is 

 {m — 7i) ax + {n —I) by -\- {I — m) cz = o, which passes through the 



centroid G (-, t, -) of the triangle ABC. Hence GG' is parallel to the 

 \a b cj 



axis of the parabola. 



From the above equations for the centroid G' ax' = k {P ~ 2tmn), 

 by' = K {nf — 2lmn), cz' = k (n^ — ^hm), and therefore ax' -f by' -f cz' 

 = — 3k Inm, and thence we obtain ax' — 2by' — Icz' = 3k P. 



The locus of G' is therefore the cubic curve 



{ax - 2by - Iczf + {by - 2cz — laxf + {cz - 2ax - 2byf = o. 



