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.

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polygon are concurrent; the locus of their point of concurrence is the Brocard circle of the polygon.
Prop. 13.-The centres of similitude of the pairs of
consecutive sides of a harmonic polygon form the vertices of
a harmonic polygon.-(TARRY.)
Dem.-Let A be a vertex of the harmonic polygon,
K its harmonic pole. Join AK, and produce to meet the
Brocard circle in M. Join MS, cutting the Brocard
circle in N, and AS, cutting the circumcircle in A'.
Join ON. Now the polar PQ of S passes through the
A' ~ d \s p
0 Sy, S
intersection of MK and ON; and since the points P, S
are harmonic conjugates to A, A', and S', S to K, 0,
the pencils Q(SKS'O), Q(SAPA'), are equal, and they
have three common rays, viz. QS, QA, QP. Hence
their fourth rays, QO, QA', coincide; therefore the
points Q, 0, A' are collinear. Again, A', being the
inverse of A with respect to Z, is a vertex of a regular
polygon inscribed in Z. Hence N is the vertex of a
regular polygon inscribed in the Brocard circle; and