SYMMEDIAN POINT OF A TRIANGLE. G'S 



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3. If any point be t:iken and the lines AO, BO, CO meet BC, 

 CA, AB in D, E, F, respectively, then the equations of the lines EP^ 

 ED, ])E are of the form 



— la + mf3 + ny =■ 

 la — w/3 + ny = 

 la + mfi — «y = 0. 

 The equation of a conic circumscribing the triangle formed by 

 these lines will be 



^ + ^ + ^—^ = 0, 



— ?a -f mp -\- ny la — m^ ■\- ny la + ni(3 — ny 



and this will be a circle if 



where 



^, = Z- + w' + n- — 2mn cos A + 2nl cos B + 2lm cos C 

 0^ — I'- + m- + n- + 2mn cos A — 2nl cos B + 2lm cos C 

 ^, = Z' + w' + «' + 2inn cos A + 2nl cos B — 2Zm cos C, 



The co-ordinates o| the symmedian point (a^y) are given by 

 — Za + in/S + ny = kX 

 la — mf3 + ny =: kfj. 

 la + m/S — ny = lev, 

 that is to say, 



IX + V V + X V + fX 



a : ^ : y = — — - : . 



I m n 



\i I : m : n ■=^ cos A : cos B : cos C, 

 then e^ = e, = e^ = 1, and 



X : fx : V ■= a cos A : b cos B : c cos C : 



hence the co-ordinates of S', the symmedian point of the pedal 

 triangle of the triangle of reference, are tan A cos (B — C) : tan B 

 cos (C — A) : tan C cos (A — B). 



The equation of the line joining S' to S (/? i c) is cos'A sin 

 (B - C) a -f cos'B sin (C — A)' y8 + cos'C sin (A - B) y = 0. 



This equation is satisfied by (sec A, sec B, sec C) : hence we 

 derive the theorem — 



"The symmedian point of a triangle, its orthocentre, and the 

 symmedian point of its pedal triangle are coUinear," 



If Z : m : n ^= a \ h : c 



then 0, : 0, : 0^ = a- : b' : c- 



= X : IX : V, 



hence the co-ordinates of S", the symmedian point of the medial 

 triangle of the triangle ABC, i.e., the triangle formed by joining the 

 middle points of the sides of that triangle, are — 



b' + c' . c' + a' . a' + b' 



