In-and-circumscribed Triangle. 21 



is therefore «c -hi 66^=0, or 



b= — \i^ ac. 



Substituting this value, the equations become 



c/iV + i \^ca{/u, + v)^ + a = 



cv2\2 + i\/ca{v + XY + a=0 



cXV + « "^caQ^ -f yLt)2 + a = 0. 



The first and second of these are 



A+2Hv + Bv2 = 

 A' + 2H'v + B'v2=0, 

 where 



A=(— i v/a + /A^ \^c)i '/a, H = i V'cap,B = {i A/a + fi^ Vc) \^c 

 A' = {-i \^a+\^ \^c)i V'a, H'=? v'c^X,B'=(z ^a-f \2 ^"^ ^^ 

 AB' + A'B - 2HH' = 2i \/«c(fl - i VacXfi + cX^^) 

 AB— H2 = i\/fl7(«— iVac/A^ + c/i") 



A'B'-H'2 = iA/^(«-r\/^X2^cX4). 



And the result of the elimination therefore is 



(a - i \^ac\^ + c\^) {a - i s^acij? + c/i^) - (« _ f v/«cX/tt + cX V)^ 

 viz. 



2 VTaQ^-ixficX^iJ? + i \/c«(X + ya)2 + «) = ; 



which agrees, as it should do, with the third equation. 

 To find the condition that it may be possible in the conic 



to inscribe an infinity of triangles, each of them circumscribed 

 about the conic 



ax^ + by"^ + cz^ = 0. 



Let the equations of the sides be 



/ Vax-^m */by + n\^cz=0 



V 's/ax-{-rn! Vby + n' \/'cz = 



I" Vax + m" Vby-\-n" \/'cz=0. 

 Then the conditions of circumscription are 

 P- +m^ +ra2 =0 



And the conditions of inscription are 



bc{mlri!> - nJ'n'f + ca(«7" - n^'l'f + ab{l'n,l' - l"m'f=0 

 bc{7n"n -mn!'f + cfl(«"/ - nl"f + ab{l"m - lm"f =0 

 bc(7n7i' -m'nf +ca{nl' -m'/)^ + ab\lm' -V mf = 0. 



