Casey — On the Harmonic Hexagon of a Triangle. 549 



Peoposttion VIII. — If K he the centre of similitude of the triangie 

 A£C (Fig. 2), and a homothetic trimigle a/?y, the circumcircle of the 

 triangle ajBy loill meet the symmedian lines AA' , BB' , CC in three new 

 points a' /S' y', which will he the vertices of a triangle homothetic with the 

 cosymmedian of the triangle AB C. 



Bern. — Since ^ is the homothetic centre of the triangles -4^6', 

 a)Sy, it is the centre of similitude of their circumcircles. Hence the 

 Hnes KA', KB', KC are divided proportionally in the points a', ^', y' ; 

 and therefore the triangles A'B'C', a'/3'y' are homothetic. Hence the 

 proposition is proved. 



Cor. 1. — The triangles a/3y, a'/3'y' are cosymmedians. 



Cor. 2. — If the sides of the triangie a/Sy produced, if necessary, 

 meet those of ABC in six points; and the sides of a'/3'y' meet the 

 sides of A'B' C in six other points, the twelve points are concyclic. 



Bern. — The first six points lie on a Tucker's circle of the triangle 

 ABC, and the other six on a corresponding Tucker's circle of A'B' C; 

 and the proposition -will be proved by showing that these circles are 

 identical. 



