330 Transactions. 



The lines joining the poniLs of contact of the conic with the sides of 

 the circumscribing triangle to the opposite vertices of that triangle are 

 concurrent in the point 



[_p + ^ — ^2 . ^2 + ^2 — P . o^ + P ~ C2 J • 



If the point 0^(a^/3^y^) lie on the circle ABC, then the locus of 

 Oi(ai^iyi) is a quartic curve whose isogonal transformation is the conic 

 a' cot A + P'^ cot B + y^ cot C - 2 (/?y sin A + ya sin B -h ay3 sin C) = o. 



If the point O^ lie on the Steiner ellipse 



_i_ 1 1 



Ti^'^ b^'^ ~cy ~ ^' 



then the locus of Oj is the quartic curve whose isogonal transformation is 

 the conic 



abc (a' + /3' + y-) - {a? + 6^ + c") {apy + hya + caji) - 0. 



§4. The remainder of the paper is concerned with the cases arising 

 when the poles of the two sub-polar triangles are isogonally conjugate 

 with respect to the triangle ABC. Let a point ©(ao/J^yJ and its 



isogonal conjugate 0'( —| be taken, and let their sub-polar triangles. 



be DEP, D'E'F. Also let 



o=— — — . = — — — , r = — — — 



^ 7o ;8o ^ ao 7o' |So ao 



, _ /3o , 7o , 7o , «o , _ tto , ^o 



7o /3o "o 7o Po «o 



The equation of the line Oo' is 



L = aa, (^o^ - y„^) + jift, (y,,^ - a„^) + yy„ (a,^ - jS^) = 0, 



which, after dividing out by ao/3oy„, reduces to 



Ij = J3a + gy8 + ry = 0. 



The sub-polar triangles lohose poles are isogonal conjugates are self- 

 conjtigate xvith respect to the conic ivhich is the isogonal transformation of 

 the line joining their poles. 



The co-ordinates of the points D, D' are respectively (OySoyo)r 

 ( g- - ), and the equations of EF and E'F' are 



a B y 



- + £ + - = 

 ao |3j 7o 



— aoo + /3^o + yyo = 0. 



The equations of the polars of D, D' with respect to the conic 



So = p^y + qya + ra(3 = 

 are 



P (^yo + y/5o) + a {qy, + r(3,) = 0, 



V (^^o + yyo) + « (2/^0 + ^yo) = o, 



which at once reduce to the equations found for EF, E'F', and so prove 

 the theorem. 



