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.

182

182 ~A SEQUEL TO EUCLID.

CA-that 'is,: DD': EE', or a3 b O. In like manner,f:y/:: b: c3-. Hence a: y:a3:'::perpendicular from r on IBC:perpendicular from r on AB.
Hence the line iBr passes through T.
Con. 3.-The per~pendiculars from the centre of perspective of ABC, pqr, on the sides of ABC are pro~portional to a3, I3, c'.
Co-R. 4,-The intersections of the antiparallel chords
D'E, E'F, F'D with Lemoine's _parallels DE', EF', FID',
respectively, are eollinear, the line of collinearity being
the polar of T with respect to Lemoine's circle.
Dem.-Let the points of intersection be P, Q, li;
then CpP forms a self-conjugate triangle with respect
to iLemoine's circle. Hence P is the pole of C~p.
Similarly Q is the pole of Aq, and R the pole of Br;
but Aq, Cp, Br are concurrent. Hence IP, Q, iR are
collinear.
Prop. 3.-ITf a triangle a3y be homothetic with ABC,
the hornothetic centre being the symmedian point of ABC;
and if the sides of a/-y produced, if necessary, meet those
of ABC in the points ID, E'; E, F'; F, ID'; these six
points are concyclic.
Dem.-Let K be the symmedian point. From the
hypothesis it is evident that the lines AK, BK, CK are
the medians of FE', IDF', ED'. Hence these lines are
antiparallel to the sides of the triangle ABC, and
therefore, as in Prop. 1, the six points are concyclic.
Con,. 1.-The circumcentre of the hexagon IDD'EE'FF'
bisects the distance between the circumcenlres of the triangles ABC, act/3.
Cont. 2.-If the triangle a/3y vary, the locus of the circumcentre of the hexagon is the line, OK.
The circumcircles of the hexagon, when the triangle
aojy varies, were first studied by M. LEMOINE at the
Congress of Lyons, 1873. Afterwards by NEUBnnG