Hogg. — On the Inscrihed Parabola. 495 



It is at once seen that the conic Sj passes through the four fixed points 



Ka'h'~c)' \ a b' cJ' W b' cj' W b' c. 



Its equation may be written 



W mm' , nn' 



+ 7-cri r- — + — : , - . , o. = o. 



2ax + by -{- cz ax + 26?/ -\- cz ax + by -\- 2cz 



Hence S,, which is the harmonic conic of S and S', circumscribes the 

 triangle Ao- It is the locus of points whose polars with respect to Aq 



(//' 1)1711^ 1171 \ 

 — , —J—, — ); ^-e., whose polars are perpen- 

 dicular to the pedal line of the point which is the trilinear pole of the 

 chord FF' of the circle ABC. Its centre lies on the conic 



by -\- cz cz -\- ax ax + by 



6. The triangle A'B'C is self-conjugate with respect to the conic 



\- -, 1 = 0, whose equation may be written in the form ll'll 4- 



ax by cz 



mm'M^ + ?m'N'- = o. This conic passes through the points G (-, t, -) 

 and P (— 7, r — . — ), and the tangents to the conic at these points are 



\(ll/ 0111/ CIX/' , 



I'ax + m'by + n'cz = o and IH'ax + vi?m'by + ■n^n'cz — o respectively ; 

 they are parallel to the axis of the parabola S. The locus of the centre 

 of this conic as F moves round the circle ABC is the maximum inscribed 

 ellipse of the triangle ABC. 



7. Let three points Fi, F.,, Fg, having trilinear ratios (t>~>~)) 



\L m n/ 



>-. t)' (-, 7, — ) respectively, be taken on the circle ABC. Th( 

 I n U \n i mj 



I m nr 

 a b c 



VI 



If* li Z 



equation of F^F^ is — , + -^ A = o : hence it touches the conic 



at bm en 



\ — I- A't+ v- = o, which is the Brocard ellipse of the triangle 

 a b c 



ABC. Hence FjFgFg is a triangle inscribed in the circle ABC and circum- 

 scribed to the Brocard ellipse of that triangle. 



Let inscribed parabolas Si, 8.3, S3 have their foci at Fi, F.,, Fg respec- 

 tively, and let Pj, P2, P3 be the centres of perspective of the triangle ABC 

 and the "contact" triangles of Sj, S2, S3 respectively with the sides of 

 the triangle ABC. The trilinear ratios of Pi, P.,, P3 are respectively 



7-, -^, — . — . -I'l-}' — . Tf. — » and the equation or P.^Pg is 

 la mb ncj \ma nb IcJ \na lb mcj 



O/OO uU GZ 



-J — 1 1 = 0, which touches the ellipse Var -f Vby + Vcz = ; 



