548 Proceedings of the Royal Irish Academy. 



Cor. 2. — The rectangle contained by the six sides of two cosym- 

 median triangles is equal to 27 times the continued product of the 

 sides of the harmonic hexagon. 



Bern. — The anharmonic ratio of the four points A, B' , C, A' is 

 equal to that of corresponding points of a regular hexagon. Hence 

 A C . £A' = 3 . A£'. CA' ; and, multiplying these and two other 

 corresponding equations, the proposition is proved. 



Cor. 3. — The continued product of the three symmedian lines of a 

 harmonic hexagon is equal to eight times the continued product of 

 three alternate sides of the hexagon. 



Peoposition YI. — If the points of intersection of the sides of two 

 cosymmedian triangles AB C, A'B' C be denoted by L, M, N, F, Q, R; 

 then the Brocard points of the triangles are isogonal conjugates, with 

 respect to the angles of intersection L, M, N, P, Q, R. 



Z)m.— Join n.4, fi^', O'^, 0'6'. Now the angle O^ J? is equal 

 to the angle Q,B'C', being, respectively, th.e Brocard angles of the 

 triangles ABC, A'B'C. Hence the four points A, L, O, B' are 

 concyclic. Hence the angle Q,IB' is equal to QAB'. In like man- 

 ner, the angle Q'ZB = d'C'B. But since the angles BCB', BAB' 

 are equal and the angles 12' C'B' and ZAfl are Brocard angles, the 

 angles O' C'B, QAB' are equal ; hence the angles Q'ZB and B'lO, 

 are equal. Hence the proposition is proved. 



Cor. 1. — The six angles of intersection of the sides of the triangles 

 ABC, A'B'C, at the points Z, M, iV, P, Q, R, are equal, respec- 

 tively, to those subtended at either Brocard point by the six sides 

 of the harmonic hexagon. 



For, since the points A, L, fi, B are concyclic, the angle ALB' 

 = A^B'. 



Cor. 2. — The angle subtended at Q. by the three alternate sides 

 BC, AB' , CA' of the harmonic hexagon are, respectively, equal 

 to those subtended at O' by the same side, taken in the order 

 CA! , BC, AB' ; and, similarly, for the other three sides. 



Cor. 3. — The feet of the perpendiculars, let fall from the points 

 li, O' on the sides of the two cosymmedian triangles ^-5 C, A'B' C , are 

 concyclic. 



Cor. 4. — If R' be the radius of the circle of Cor. 3, R' =^ R sin w. 



Peoposition VII. — If in any triangle ABC a triangle similar to it<i 

 cosymmedian be inscribed, the centre of similitude of the inscribed tri- 

 angles is the symmedian point of the original triangle ABC. 



Bern. — Let K be the symmedian point ; then the angle BKC is 

 equal to the sum of the angles BAC, ABK, KCA; that is, equal 

 to the sum of the angles BAC. B'A'A, AA' C ; or the sum of 

 BAC, B'A'C. Hence {Sequel, Book III. Prop. 17) the propositon 

 is proved. 



