492 Transactions. 



The equation of the pedal line of F is 



Imn 2E sin A sin B sin C {ax -\- by + cz) 



= {mna^ + nlh^ + Imc-) (I cos Ax + m cos By + n cos C^;) ; 



hence, eliminating I, m, ?i between this and the equations, I -\- m -\- n = o, 

 {m — n) ax + {11 — l)hy-\-{l — m) cz — 0, we obtain for the pedal of the 

 envelope of the pedal lines of the triangle ABC, the eentroid being the 

 pole of the pedal, the equation 



2B sin A sin B sin C 2 {ax) UVW = 2 (a^VW) 2 (cos A icU), 



where JJ ^ by -\- cz — 2ax, Y ^cz -\- ax — 2by, 'W = ax -\- by — 2cz. 



3. A parabola is the isogonal transformation with respect to any 

 inscribed triangle of a tangent to the circum circle of the triangle. 

 Hence the parabola S = 0, whose equation may be written in the form 



=- + ^Tf + XT = o> '^^V ^^ regarded as the isogonal transformation with 



L M N ^ 



respect to the triangle LMN of a tangent to the circumcircle of that 

 triangle. 



The equation of this circle is 



where Oi = a'^P + b'^m^ + c-ii^ — 2bcmn cos A + Icanl cos B + 2ablm cos C, 

 with similar expressions for Q^ and Q3.* 



The tangent to this circle at the point Q, determined on it by the 

 I'll m'M w'N . i'^h ra'^M n'''N 

 equations ^^ = ^^ = :^, is J^^ + ^^^+^^^ = 0, 



where V -\- m' -\- n' = . 



The isogonal transformation of this line with respect to the triangle 



^'2 ^jj'2 ^1^1% 111 



LMN is ^5^ H ^^ -|- — ^ = 0, which reduces \,o -=- -\- ^4- ^^ — if 



in . ^y^i2 . ,j'2 _ p . ^^2 . j^2_ Hence S is the isogonal transformation of the 

 tangent -— + — + — = 0, whose point of contact Q is given by the 



IZ1 4^2 ^^3 



e nations - - A _ ^ 



To determine the locus of Q we have k (M -f- N) = mU.^ + ni\, 

 whence 2k lax = {m + n) {l^a^ + vi^b^ + w'^c^ + %nnbc cos A) 

 + 2Z {ni^ab cos C + ifca cos B) — 2lmna^, 

 2k ax = — (1%^ + ^^^^ + ^^c^ + 2mnbc cos A) 



+ 2 (??i%6 cos C + n'^ca cos B) — 2mna^ 

 = I {m + n) a~ + m (n + Z) ¥ -\- n {I + m) c- — 2nmbc cos A 

 — 2m {n + Z) a6 cos C — 2n {I + m) ca cos B — 2vma^ 

 = — 3mna^ + wZ6^ + Z?«c^ 



* See " Messenger of Mathematics," No. 502, February, 1913, p. 129. 



