Casey — On the Harmonic Hexagon of a Triangle. 545 



XXX. — On the Haemonic Hexagon oe a Triangle. By John 

 Caset, LLD., F.R.S. 



[Read, January 26, 1886.] 



Definition I. — If ABC be any triangle; AA', BB', CC its 

 symmedian lines, produced to meet its circumcircle in the points 

 A', B', C, the hexagon, whose vertices are the six points A, B', C, 

 A', B, C\ possessing several geometrical properties, it is convenient 

 to have a definite name for it. I propose to call it the ha/rmonic 

 hexagon of the triangle. 



Definition II. — Two triangles, having the same symmedian lines, 

 are called cosymmedians. 



Definition III. — The lines AA' , BB', CC are called the syni' 

 median lines of the hexagon. 



Proposition I. — The triangles ABC, A'B'C are cosymmedians. 

 (Fig- 1)- 



