HocG. — Theorems relating to Suh-Folar Triangles. 329 



Hence the theorem, — 



A rectangular hyperbola can be drawn through the middle points of the 

 Sides of a triangle and the ■points in wJiich tJie sides of the triangle are cut 

 by lines joining any point on the circum-circle of the triangle to the vertices 

 of the triangle. 



11 Oi be the orthocentre of the triangle ABC, then conic (iv) re- 

 duces to 



cos(B-C) cos(C-A) co«(A-B) 



! -I '-^ \- ! -' = 0. 



§ S. If the circle described about the tiiangle DiEjFi meet the sides 

 BC, CA, AB of the triangle ABC again in the respective points DiE^F\ 

 then the lines AD\ BE\ CF^ are concurrent. Let the co-ordinates of 

 tlio point of concurrence O^ be (a^/S^y^). The vertices of the triangles 

 DiEiFi, D^E^F^ he on the circle 



^mJ-u. ''' _ fi (-1- _i_ J_^ (^- -1. ^ ^ 



From the conditions for a circle, we at once obtain 



^ : -^ : — 1 = a (yi'af + a.^/S^- - /Sfyi^) - 2ai/3jyi cos A{aa, + b/3, + Cyi) 

 : b {a,-f3r + ftfyi^ - jiW) - 2ai/3iyi cos B (aai + b/3i + Cyi) 

 : C (/?/yi^ + yi'«i'^ - a//5i"^) - 2ai^iyi cos C (aai -I- b(3i -t- Cyi). 



It may be at once verified that if aj : /3i : yj = ^ ' b ' '' ^^^^ 

 ■a^ : (3^ : y^ = sec A : sec B : sec C, and the circle in this case is the 

 nine-point circle of the triangle ABC, 



If aa^ + b/Si -f Cyi = 0, then 



W^ + cyj 



a,^ 



hence the locus of the point a^/^y is the quartic curve 



This result may be stated as follows : — 



If lines drawn through the vertices of the triangle ABC parallel to a 

 given line L meet the sides of that triangle in DjEiFi, and the circle 

 through DjEiFi intersect the sides again in D^E^FS then the lines ADS 

 BE\ CF^ are concurrent in the point 0\ and as L turns about a fixed 

 point in the plane ABC, the locus of 0^ will be the above curve (vi). 

 This quartic curve is the isogonal transformation of 



Va{by -f c/3) + Vb{ca + ay) -f Vc (a/3 + ba) = o, 



a conic inscribed in the triangle formed by drawing tangents to the circle 

 ABC at the vertices of the triangle of reference. 



