546 Proceedings of the Royal Irish Academy. 



Bern. — Since the three lines AA' , BB', CC are concurrent, the 

 six points in which they intersect the circle are in inyolution. 

 Hence the anharmonic ratio of the four points B, A, C, A' is equal to 

 tho anharmonic ratio of their four conjugates B' , A' , C, A ; but since 

 A A' is a symmedian of the triangle ABC, the four points B, A, C, A' 

 form a harmonic system. Hence the four points B', A', C, A form 

 a harmonic system. Therefore AA' is a symmedian of the triangle 

 J'B'C. Similarly, BB', CC are symmedians, and therefore the 

 triangles are cosymmedians. 



Proposition II. — If two triangles he cosymmedians, the sides of one 

 are proportional to the medians of the other. 



Bern. — The angle B'A'C is equal to the sum of the angles B'A'A, 

 AA'C: that is, equal to the sum of B'BA, ACC; and these angles 

 are, respectively, equal to the angles which the medians from B and 

 C of the triangle ABC make with BC. Hence, if G be the centroid 

 olABC, the angle ^^C is the supplement oiBGC; and therefore 

 the angles of the triangle A'B'C are equal to the angles of a tri- 

 angle whose sides are the medians of ABC. Hence the proposition 

 is proved. 



Proposition III. — lemoine's First Circles of two cosymmedian tri- 

 angles are identical. 



Bern. — Let be the common circumcentre of the triangles ; R its 

 radius ; p, pi, the radii of their first Lemoine Circles ; then, by a 

 well-known property of these circles, 



Hence p = pi ; and since the middle point of OK'x^ the centre of each, 

 the circles are identical. 



Cor. 1 . — lemoine's Seco7id Circles of two cosymmedian trianglos are 

 identical. 



For if p', p/ denote the radii of their Lemoine's second circles, 

 we have 



p'^ = P^-(^J=Pi'^ Hence p' = px'. 



Cor. 2. — The Brocard angles of two cosymmedian triangles are equal. 

 Eor if the Brocard angles be co, w', we have 



1-3 tan^w = ■„, =1-3 tan- w'. Hence w = co'. 



Cor. 3. — The Brocard points of a triangle are also the Brocard 

 points of its cosymmedian. 



