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.

220

A SEQUEL TO EUCLID.

therefore M, which is its inverse, is a vertex of a harmonic polygon; but M is evidently the middle point of
the symmedian chord passing through A. Hence it is
the centre of similitude of the two consecutive sides
passing through A. Hence the proposition is proved.
Prop. 14.-If figures directly similar be described on
the sides of a harmonic polygon of any number of sides,
the symmedian lines of the harmonic polygon formed by
corresponding lines of these figures pass through the middle
points of the symmedian chords of the original figures.
This is an extension of Prop. 5, page 197, and may
be proved exactly in the same way.
COR. I.-The symmedianpoint of the harmonic polygon,
formed by corresponding lines of figures directly similar,
is a point on the Brocard circle of the original polygon.
CoR. 2.-The invariable points of the original polygon
are corresponding points of figures directly similar described on its sides.
CoR. 3.-The centre of similitude of the original polygon, and that formed by any system of corresponding lines,
is a point on the Brocard circle of the original polygon.
COR. 4.-The centre of similitude of any two harmonic
polygons, whose sides respectively are two sets of corresponding lines of figures directly similar, described on the
sides of the original polygon, is a point on the Brocard
circle of the original.
Exercises.
1. If the symmedian lines through the vertices A, B, C of a
triangle meet its circumcircle in the points A', B', C', the triangles
ABC, A'B'C' are cosymmedian.
For since the lines AA', BB', CC' are concurrent, the six
points in which they meet the circle are in involution. Hence
the anharmonic ratio (BACA') = (B'A'C'A); but the first ratio is
harmonic, therefore the second is harmonic. Hence A'A is a