A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

THEORY OF HARMONIC POLYGONS. 209
Prop. 3.-If 8 be the distance of the symmedian point
K from the circumcentre, and R the circumradius of the
polygon, tan o = //(1 - jR2) cot 7r/n.
Dem.-We have Def. vi. (K, AB)/ AB = 2 tan o;
and since the polygon whose side is A'B' is regular, and
has n sides (0, A'B')/A'B' = 2 cot w-/n. Hence tan w:
cot Tr/n:: SK: SO. Again, since the points 0, K
are harmonic conjugates to S, S', and S, S' are inverse
points with respect to Z, it is easy to see that
SK/SO = /(1 _-S2i2)
Hence, tan o = V/(1 - 8R2) cot wr/n.
COR. 1. —2 = R2 (1 - tan2 (o tan2 rr/n).
COR. 2. —If two harmonic polygons of m and n sides
respectively have a common circumcircle and symmedian
point, the tangents of their Brocard angles are:: cot 7r/m: cot r /n.
COR. 3.-Since the side A'B' of a regular polygon of n
sides may have any arbitrary position as a chord in the
circle, it follows that an indefnite number of harmonic
polygons of n sides, and having a common symmedian
point, can be inscribed in the circle.
CoR. 4.-The anharmonic ratio of any four consecutive
vertices of a harmonic polygon is constant.
Prop. 4.-If Ao, A... A,_ be the vertices of a
harmonic polygon of n sides, the chords A1Al, A2A,,2,
are concurrent.
Dem.-Let K be the symmedian point. Join AoK,
and produce to meet the circumcircle again in Ao'.
Then the points Ao, Ao' are harmonic conjugates with
respect to the points Al, A,,_1 (Demonstration of Prop. 1).