Casey — On the Harmonic Hexagon of a Triangle. 555 



Dem. — Let us consider the trianoie formed by the three sides 

 BC, A£', CA'. Let them meet in" the points A"', C", £'"; it is 

 required to prove that the triangle A"'B"' C" is in perspective with 

 the triangle NL Q, formed by the invariable points corresponding to 

 the sides £' C, A'B, C'A. Join the points N, L, Q, respectively, to 

 £", and produce to meet the sides BC, AB', CA'. They will meet 

 them in points where the same sides are intersected by Lemoine's first 

 hexagon circle ; and since Lemoine's first circle and the Brocard circle 

 are concentric, the parts intercepted by them on the line EN will be 

 equal. Hence the lines A!"N, A!"K are isotomie conjugates with 

 respect to the angle A!" . Similarly, B"'Q, B"'K are isotomie conju- 

 gates with respect to the angle B'" ; and C"'L, C"'K with respect 

 to the angle C". Hence the lines A'"]^, B"'Q, C"'L are concur- 

 rent. Q.E.D. 



NOTE ADDED IN THE PRESS. 



Since writing the preceding Paper, I have succeeded in showing 

 that the propositions contained in it are capable of remarkable exten- 

 sions. I do this by proving that we can construct a harmonic polygon 

 of any number of sides that is a cyclic polygon, having a point in its 

 plane called its symmedian point, such that perpendiculars from it on 

 the sides of the polygon are proportional to the sides. The solution 

 of this problem is contained in the following theorem : — 



The inverses of the angular points of a regular polygon of any numher 

 of sides form the angular points of a harmonic polygon of the same numher 

 of sides. 



Bern. — Let A^ B, C, &c., be the angular points of the original 

 polygon ; A', B', C the points diametrically opposite to them. Now 

 invert from any arbitrary point. The circumcircle of the original 

 polygon will invert into a circle, and the lines AA' , BB', CC", &c., 

 into a coaxal system, and the radical axes of each circle of this system 

 and the inverse of the circumcircle will be a concurrent system of 

 lines (^Sequel, Book YI., Sect, v., Prop. 4). Now if the inverses of 

 the points A, B, C, &c. ; A', B', C, &c., be a, P, y, &c. ; a', (3', y', 

 &e. ; the concurrent lines will be aa', ySyS', yy', &c., respectively ; let 

 E be their point of intersection. Now, since evidently the pionts 

 A, B, C, B' form a harmonic system, the points a, /?, y, P' form 

 a harmonic system. Hence the perpendiculars from the point K in 

 Pfi' on the lines a/3, /3y are proportional to these lines. Hence the 

 proposition is proved. 



It is evident now that the whole theory of Brocard circles, Brocard 

 points, Lemoine circles, cosine circles, similar figures, invariable 

 points, double points, (S:c., can bo extended to harmonic polygons 

 of any number of sides. The following are a few of the numerous 



R.I. A. PROC, SER. 11., VOL. IV. SCIENCE. 3B 



