STMMEDIAX POINT OP A. TRIAXGLE. 



65 



Hence since the Broeard ellipse of any triangle is the envelope of 

 the axes of homology with respect to that triangle of points lying on 

 the axis of homology of the symmedian point of the triangle, the 

 Broeard ellipse of a triangle may be at once found from the conic C, 

 when the co-ordinates of the symmedian point are known. Thus the 

 Broeard ellipses of the pedal and medial triangles of the triangle of 

 reference are respectively — 



V- 



— a COR A + /8 COS B + y Cos C 



sin 2A 



+ 



Va COS A — /S COS B + ycosC 

 shi2B 



+ 



/a cos A + Z? cos B — y cos C 

 V siQ 2C 



= 0. 



- \/ — aa -\- bp + cy + V «a — bft + cy -V - \/ aa + bfS —cy — Q. 

 a be' 



The Broeard ellipse of the triangle IJJ- is — 



6. "If any two mutually perpendicular lines be drawn through 

 the symmedian poiat of a triangle ABC, then the line joiaing the two 

 points on the circle ABC, of which these lines are the axes, passes 

 through a fixed point collinear with the centroid and orthocentre of the 

 triangle ABC." 



Let any two points, P' (a'/3'y') and P'' (tt"/3''y"), be taken on theTf) /" 

 circle ABC. The axes of P' and P" are— ,< A ^" 



a' ^' ^ y' 



a"+ /3"^ y" 

 and these lines are at ri^-ht ano-les to each other if — 



, „ — cos A I , „ + -f^^-j I 



n \py Py / 



- cos B ( ) „ + ; , ) - cos C ( \, + -^) = 0. 

 Vy.i. ya/ \ap a /3 / 



It may be easily shown that — • 



h /A_ 4. J_\- "' _ ^'' _ ^' 



*H/3y ^ f3"y')~ -V' fi'/3' yV 

 with similar relations for the coefficients of cos 1? and cos C. 



On substituting in the above condition of perpendicularity we 

 obtain — 



-}~j, (1 + 2 sin'A cot B cot v] + J,,,, J 1 + 2 sin'B cot C cot A j 



1 1 



' " + R'rr + 



a u p p y y 



+ } „ ( 1 + 2 sin'C cot A cot B j = 0, 



