Casey — On the Harmonic Hexagon of a Triangle. 553 



fore O is a point on the Brocard circle. Similarly the lines CiV, B'P^ 

 A Q, C'R each intersect BL on the Brocard circle, and therefore each 

 passes through Q,. In the same manner it may be shown that the 

 sis lines A'L, CM, B'N, AP, C'Q, BR are concurrent, and meet in 

 another point O' on the Brocard circle. 



Def. — fi, O' are called the Brocard points ; and L, M, N, P, Q, R 

 the invariable points of the hexagon. 



Cor. 1. — O, O' are isogonal conjugates with respect to each angle 

 of the harmonic hexagon AB'CA'BC. 



For the angles A'BQ, C'BQ,' are each equal to the Brocard angle 

 of the hexagon. Hence, &c. 



Cor. 2. — The feet of the perpendiculars from fl, O' on the sides of 

 the hexagon are concyclic. 



Cor. 3. — If w be the Brocard angle of the triangle ABC, and & 

 the Brocard angle of the hexagon, tan = 3 tan co. 



