64 PROCEEDINGS OP SECTION A. 



The equation of the line joining SS" is — 



a (Jr — C-) a + b (fi- — a') (3 + c {a- — h') y = ; 



this being satisfied by ( - , , I, we derive the theorem — 



\ a 'b G / 



" The symmedian point and centroid of a triangle are collinear 

 with the symmedian point of its medial triangle." 



The equations of the nine-point circle of the triangle ABC 

 corresponding to these two cases may be written — 



a cos A _i_ ^ ^^^ ^ 



a cos A + yg cos B -f y cos a cos A — /3 cos B + 7 cos C 



, c cos C 

 + ^ 0^ 



a cos A + /3 cos B — y cos C 

 a- I ^' , c- 



+ —-- — + ~ = 0. 



— aa + b(3 + cy aa — bfS + cy aa + b/3 — cy 



4. If three points (Xa, fxb, fo-), (/xr/, vb, Ac), (m, \b, /xc), where 

 A+/^ + i'=0, be taken on the line — ■ • 



" + f + -r = 0. 



a 



the axes of homology of these points form a triangle A^B^C^ inscribed 

 in^the circle ABC and circumscribed to Brocard's ellipse, 



The axes of homology of the triangle ABC will intersect at the 

 symmedian point of the triangle A,B,Ci. 



Solving for the point of intersection of the axes of B and C, we 

 obtain — 



^ : ^ : "^ = 9 X'ljrir -0 :e:—:^ XavO, : 6:' - :^ Xuve,, 

 a b c ' i-2 r-ii f«j 



where 0^ = A/x' + /xr' -f I'A' 



Oj ^= A'/x + /x'r + v'X. 

 Hence, observing that A' — /^if=/x' — iv\ = i'' — A/x = ^, 



we find "^ = f = ^'' =^. 



a b c 



Hence the theorem — 



"All triangles inscribed in the circle ABC and circumscribed to 



the Brocard ellipse of the triangle ABC have a common symmedian 



point." 



5. The envelope of the axes of homology with respect to the 

 triangle L' L" L'" (section i.) of points lying on the axis of homology 

 of the point (ao/J^yo) with respect to that triangle is the conic 



Vz!' +Vi>a/^]"=» ■■■■'■' 



