Hogg. — On Ceniroiddl Triangles. 611 



10. The equations of the sides of the secondary centroidal triangle 

 PQR formed by joining the points of contact of L', M', N' with S'S"S"' 

 respectively are 



SpY aa + (i)* + 'ipif) b/3 + iq' + "Ip'q) cy - o 



and two others. 



Writing A = - we have 



X'hfB + 2A'cy + 3A'^aa + 2A^/3 + cy = o. 



The invariants I and J of this quartic are 



I = !f!^, J = J-(6a6cay8y - 26«^^' - 2cV - aV), 



4 o 



whence the envelope of the above line is 



(a«a« + b'lS' + C'V - 3abcal3y){h'l3' + cy - 3a6ca/3y) = 0, 



whence we may infer that the envelope of the sides of the triangle PQR 



consists of the line at infinity, the point (-, j, -], and the system of cubics 



b'/S'' + c'V - Sabc a/3y = o 



C' y-* + <fa!^ — Sabc a/3y = 



aV + b'/3^ - Sabc af3y = o. 



11. The four common tangents of the conic S' = aV — 4:bc(3y = o and 

 the circle S = af^y + bya + caj3 = o form a cyclic quadrilateral. 



The equation of the locus of the pole of the hne -X'cy + Xaa + bj3 = o 

 with respect to the circle S is the conic 



be {c(3 + by)' — a^ (ay + ca){bl3 + aa) = o, 

 which may be written 



a^S + be IJ.^ = 



where Zj = aa + c^ + by = o 



= L-ib-c)l/3-y) = 



/j = «a — C/3 — 6y = O 



= L- [b + c)((3 + y) = 0, 



L being the line at infinity. The first forms of l^ and l^ show that they 

 each pass through the point T in which the tangent to the circle ABC at 

 A meets BC. The second form shown that l^ and l^ are parallel respec- 

 tively to the internal and external bisectors of the angle A. 



The tangents to S at the points in which it is met by l^ and l^ are the 

 common tangents of S and S'. 



The line l^ will always meet the circle S in real points ; the ine l^ 

 will meet it in real points if «^ > Abe. 



If two chords of a circle are at right angles, the tangents at their 

 extremities form a cyclic quadrilateral ; hence since l^ and l^ are at right 

 angles to each other, it follows that the common tangents to S and S' 

 form a cyclic quadrilateral which is real if a^ > Abe. 



If a^ = 4:bc, the parabola S' and the circle S touch each other, the line 

 /j being the tangent at the point of contact. 



The equations of the four common tangents of S and S' are 



[a {a^ - 4:bc) S + be l^'] [a (a^ + 46c) S + bcl^] = o. 



