Hogg. — Theorems relatiny to Suh-Polar Triangles. 331 



The vertices of the triangles DEF, D'E'F' lie on the conic 

 S = a^ + ;Q2 + y' - v'Py - g'ya - r'ayS = o, 

 and the sides of the two triangles touch the conic 



i>p qq' rr' 



The polars of the points O, O' with respect to the triangle ABC are 



a B y 



— + ~ + ~ = o, 



"o /3o 7o 



aa^ + /3/3o + yy„ = 0, 



and these lines meet in the point U whose co-ordinates are (p q r). 



Let the sides EF, FD, DF, E'F', F'D', D'E' touch 2 in the points 

 P, Q, R, P', Q', E' respectively : the equations of PP', QQ', RR' are 



Hence the triangle formed by PP', QQ', RR' is the sub-polar triangle of 

 the point W [p'q'r'). 



The conic S is the harmonic conic of 



«o ^o 7o 



2^ = Va^a + v/^,^ + V:^y = 0. 



The co-ordinates of intersection of the two conies 2i, ^2 are 



(y _ 2, r/- 2, r'- 2) (y- 2, q'+% r'+ 2) 



{p'+ 2, q'-% r'-f- 2) (p'+ 2, r/+ 2, r'- 2). 



The lines joining these points ai"e of the form 



a{q'-y')-{v'-1){li-y)=o, 



a(5'+r')- (p'+2)(^-fy) = o, 



and it is easily shown that the co-ordinates of the intersections of the 

 joining lines not lying on 2i and %, are {— p q r) (p — q r) {p q — r). 

 These are the points in which the corresponding sides of the triangles 

 DEF, D'E'F' meet. Hence the theorem, — 



The intersections of cor resimn ding sides of tivo sub-polar triangles 

 whose poles are isogonal conjugates determine the vertices of the diagonal 

 triangle of the quadrangle formed by the intersections of the tivo conies in- 

 scribed in the triajigle of reference lohich are the envelop)es of the polars 

 tvith respect to ticat triangle of poitits lying on the polars of the pioles of 

 the tivo sub-polar triangles. 



* "Messenger of Mathematics," No. 451, November, 1908, p. 117. 



