588 Transactions. 



If P be one of the two points in the plane of the triangle ABC whose 

 pedal triangles are equilateral, then AP sin A = BP sin B = CP sin C ; 

 hence the points determined by the above equations have their pedal 

 triangles equilateral. It is at once seen that this pair of points lies on 

 the Brocard diameter whose tripolar equation is 



a 2 (b 2 - c 2 ) X + 6 2 (c 2 - a 2 ) Y + c 2 (a 2 - b 2 ) Z = o. 



We now proceed to find the equation of the circle cutting the nine- 

 point circle of the triangle ABC orthogonally. 



Let D, E, F be the middle points of BC, CA, AB respectively, and 

 let X a = PD 2 , Yj = PE 2 , Zj = PF 2 , where P is a point whose tripolar 

 co-ordinates are (X, Y, Z), then 



2X X = Y + Z - | 2 



b 2 

 2Y, = Z + X - - 



SZ, = X + Y - | 2 



Let U — — aa. + fy3 + cy — o, V = aa — b(3 + cy = o, W e = aa + b(S 

 — cy = o, be the equations of the sides of the triangle DEF, and let 

 (a /3 y ) be the co-ordinates referred to the triangle ABC and (^(3^) the 

 co-ordinates referred to the triangle DEF of the centre of a circle cutting 

 the nine-point circle orthogonally, then 



act! = U , 6ft = V , cyi - W . 



The equation of the circle referred to the triangle DEF is 



aajXj + b(3 1 Y 1 + c yi Z x = o, 

 and this, when referred to the triangle ABC, becomes 



U [y + Z-| 2 ]+Y [z + X-| 2 ] +W [x+ Y- - 2 ] =o; 



*■*■' aa Q X + b(3 Q Y + c 7o Z - \ (a 2 U + b^ Q + c 2 W ), 



which reduces to 



aa Q X + 6/? Y + c 7o Z = 2R a (a cos A + /3 C> cos B + y cos C). 



The envelope of circles having their centres on the circle ABC and 

 cutting the nine-point circle of the triangle ABC orthogonally is 



«(X --^bc cos A ] + hi Y - ^ca cos Bj" + c ( Z - ^ab cos Cj = o. 



If circles be described cutting the nine-point circle orthogonally and 

 having their centres at the vertices of a triangle inscribed in the circle 

 ABC and circumscribed to the Brocard ellipse of the triangle ABC, their 

 points of intersection will lie on the curve 



+ -^^ ,+^-^ t-o. 



a?(x - | be cos A\ b* (y - ~ ca cos b\ c 2 (z - I ab cos c) 



