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.

180

A SEQUEL TO EUCLID.

is antiparallel to BC. Similarly DF' is antiparallel to
AC. Hence the angles AFE', BF'D are equal; hence
it is easy to see that FE' is equal to F'D. In like
manner it is equal to ED'.
Again, if O be the circumcentre, OA is perpendicular to FE'; therefore the perpendicular to FE', at its
middle point, passes through the middle point of KO.
Hence, since FE'= ED'= DF', the middle point of KO is
equally distant from the six points F, E', E, D', D, F.
This proposition was first published in 1873, at the
Congress of Lyons, Association Francaise pour l'avancement des Sciences, by M. Lemoine, who may be regarded as the founder of the modern Geometry of the
triangle. It was rediscovered in 1883 by Mr. Tucker,
Quarterly Journal of Pure and Applied Mlathenatics,
p. 340.
I proved, in January, 1886 (Proceedings of the Royal
Irish Academy), that polygons of any number of sides
called harmonic polygons, can be constructed, for which
a corresponding proposition is true. [See Section vI.]
DEF. —We shall call the circle through the six points
F, E', E, D', D, F' Lemoine's circle, and the hexagon of
which they are the angular points Lemoine's hexagon.
COR. 1.-The sides of the triangle ABC are divided
symmetrically by Lemoine's circle.
For it is easy to see that
AF: FF':: F'B: ab2: 2;
BD: DD': D'C:c2: a2: b2;
CE: EE':: E'A: a2: b2: c2.
COR. 2.-The intercepts DD', EE', FF' are proportional to a3, b3, c3.
Dem. —Let fall the perpendicular AL; then, since
the triangles DKD', BAC are similar,
DD': x:: BC: AL:: a2: 2A.