In-and-ch'cumscribed Triangle. 25 



and for the equation of a-, 



Then if {x\, v/„ z^), {x^, ij^, z^) are the coordinates of the points 

 V, V respectively, we have 



And the condition in order that the chord may touch the conic 

 o- is 



bc{yx-2-y'i^\f + ca[::^Xc^-z.2^]Y + ab[x^y^-x^y^f = Q. 

 But we have 



= [xyXci, + 2/1^2 + -i^a) G-^i^a-^iys- "1-2) • 

 And making the hke change in the analogous quantities, and 

 putting for shortness 



«= —bc + ca + ab 



^= bc — ca + ab 

 y— bc-{-ca — ab, 



the condition in question becomes 



(^1^2 + 2/1^2 + 2'i^2) («*i*2 + /5yiy2 + y^i^a) = O- 



But the equation a;ir2 + 2/ iyg + ^i3'2 = must be rejected, as giving 

 with the equations x^^ -Hyj^^. ^^2^0, 3^22 + ^2^-2^ = the rela- 

 tion 0,^1 : yi : ^i = A'2 •• 2/2 : v^> ^e have therefore 



which implies that the points {x^,y^,z^) and {x^, y^, ^2) are har- 

 monics with respect to the conic 



ctx^ + ^y^ + yz^ = 0, 



which is a conic having with S, a, a system of common conjugate 

 points. The equation may also be written 



{-bc + ca + ab)x'^+{bc--ca + ab)y^ + {bc + ca-ab)z'^ = 0; 

 or, as it may also be written, 



{be + c« + ab) {x^ + y'' + z^) -2[bcx^ + caz"^ + abx"^) . 

 And, as before remarked, bcx^ + cay^ + abz^ =0 is the equation 

 of the conic which is the polar of ax'' + hf + cz^=Q with respect 

 tOAVy' + ^' = 0. .^ , . ^ 



The condition in order that there may be inscribed m the 

 conic a.'2 + v/"^ + r'^ = an infinity of triangles the angles of which 



