332 Transactions. 



The sides of the triangles PQR, P'Q'E,' pass through the vertices of 

 the triangle ABC, the equations of QR and Q'R' being respectively 



qq rr 



The triangles PQR, P'Q'R' are self-conjugate with respect to the 

 follow^ing inscribed conic : — 



Va + V4 + V^ = o. 



Any point on the conic 



So = tto /?y + fto ya + To a/5 = 



is expressed by the co-ordinates 



where k is a variable parameter. 



The equations of the sides EF, FD, DE of the triangle DEF are 

 respectively 



«A(- + v)+y= (" + -) = ". 





Hence EF passes through the point of intersection of the tangents to S^, 

 at B and C; FD passes through the point in v^^hich the line joining B to 

 the pole of the conic meets the tangent at C ; DE passes through the 

 point in which the line joining C to the pole of the conic meets the 

 tangent at B. 



The sides of the triangles D'E'F' are respectively 



K^/?„y + X (ao« + /3o^ + yoy) + ^Jo = O, 

 x^'l^oY - X («oa + Poft - lol) - f^Jo = O, 

 x^/^oY + X (a^a - /3„^ + y^y) ~ /3y^ = 0, 



and these lines envelop respectively the conies 



(a,a + (3,fi + y,y)-^ - 4/3,y„y8y = 0, 

 (a^a + ^^(3 - y^y)-^ + 4/3„y„^y = 0, 

 (a,a - (3,/3 + y,y)-^ + 4^,y,/3y = 0. 



Hence the theorem, — 



If the 2^ole of a s2tb-2Jolar triangle move on a conic circumscribing the 

 triangle of reference, the sides of the suh-polar triangle ivill pass throuyJi 

 three fixed points, luhile if the pole move on. a straight line the sides of the 

 sub-polar triangle will envelop three fixed conies. 



