Casey — On the Harmonic Hexagon of a Triangle. 551 



PkopositiojS' X. — If through the symmedian point of a harmonic 

 hexagon parallels he draicu to the tangents at its vertices; then taking 

 the points in which each parallel meets the sides of the hexagon passing 

 through the corresponding vertex, the twehe points of intersection are 

 xioncyclic. 



Bern,. — Let Khe the symmedian point of the triangle ABC; then 

 BA', A' C are two sides of the harmonic hexagon. Through K draw 

 ^Z7paraliel to A'T, then ZZwdll be one of the twelve points of inter- 

 section. Draw Xr parallel to AT. Xow the angle ^KT^ TA' TI 

 = A'AV. Similarly the angle -4 F^= ^^'C^. Hence the triangh-s 

 ^XF, ^ZZ"^' are equiangular. 'Kence KU . KV = AK . KA' . I^ow 

 KVi?, the radius of the second Lemoine circle, both for the trian^de 

 ABC and its cosymmedian. Hence KU has the same value for each 

 of the twelve points of intersection, and therefore the twelve inter- 

 sections are concyclie. 



Def. — We shall, from its analogy to the case of the triangle, call the 

 circle through these twelve points Lemoine'' s second hexagon circle. 



Cor. 1. — The intercepts which Lemoine' s second hexagon circle 

 make on the sides of the hexagon are proportional to the cosines 

 of the angles which these sides subtend in the circumcircle. Hence 

 it may be called the cosine circle of the hexagon. 



Cor. 2. — If the sides of the harmonic hexagon a/3'ya'y3y' (see fig., 

 Prop, vm.), be produced to meet the sides of AB' CA' B C , viz., each 

 side of the former intersecting the two sides adjacent to the side 

 parallel to it in the latter, the twelve points of intersection are 

 concyclie. 



If R, B! be the circumradii of the hexagon AB' CA'BC, 

 af^'ya'Py', and R" the radius of the cosine circle of AB' CA'B C ; 

 then it may be proved, as Prop. vm. Cor. 2, that the twelve points of 

 intersection lie on a circle, the square of whose radius is 



'R + R' :■ ^.. fR-R'\2 

 t- R 



\ 2 J \ 2R 



