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.

234

A SEQUEL TO EUCLID.

21. Brocard's second quadrilateral is a harmonic quadrilateral.
22. If o be the Brocard angle of a harmonic quadrilateral
ABCD, cosec2 o - cosec2 A + cosec2 B = cosec2 C + cosec2 D.
23. If the middle point F of the diagonal AC of a harmonic
quadrilateral be joined to the intersection K' of the opposite
sides AB, CD, the angle AFK' is equal to the Brocard
angle. (NEUBERG.)
24. The line joining the middle point of any side of a harmonic quadrilateral to the middle point of the perpendicular on
that side, from the point of intersection of its adjacent sides,
passes through its symmedian point.
25. If F1, F2, F3... be figures directly similar described on
the sides of a harmonic polygon ABC... of any number of sides,
and if aly... be corresponding lines of these figures; then if
any three of the lines ay?... be concurrent, they are all concurrent.
26. In the same case, if the figure ABC... be of an even
number of sides, the middle points of the symmedian chords of
the harmonic polygon aBy... coincide with the middle points
of the symedian chords of ABC.
27. If F1, F2, F3 be three figures directly similar, and B1, B2,
B3 three corresponding points of these figures; then if the ratio
of two of the sides BB2: B2B3 of the triangle formed by these
points be given, the locus of each is a circle; and if the ratio
be varied, the circles form two coaxal systems.
Dem.-Let Si, S2, S3 be the double points: then the triangles
S3B1B2, S1B2B3 are given in species. Hence the ratios BiB2:
S3B2, and B2B3: SiB2 are given; and the ratio B1B2: B2B3 is
given by hypothesis. Hence the ratio S3B2: S1B2 is given, and
therefore the locus of B2 is a circle.
28. If through the symmedian point K of a harmonic polygon
of n sides be drawn a parallel to any side of the polygon, intersecting the adjacent sides in the points X, X', and the circumcircle in Y, Y', then 4XK. KX' sin2 - = YK. KY'.
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