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.

212

212 ~A SEQUEL TO EUCLID.

Prop. 7.-If the perpendiculars from the vertices of a
harmonic polygon on its Lemoine line be denoted by p,, P2
p,,, and the perpendiculars on it from the Lemoine
point by 7r, then Y, (l/p) = n/-n-.
Dem.-Let LL' (fig., Prop. 2) be the Lemoine line.
Then, since the points 0, 5', K, S form a harmonic
system 1W/OS', 1W/OK', 1W/OS are in AP [Book 'VI.,
Sect. iii., Cor.]; that is, OS, OQ, OS are in AlP.
Therefore S'S is bisected in Q. Hence VS is bisected
inY; and since the points A', A are harmonic conjugates
to VY 5, the lines UA, ITV, lTlA' are in GP [Book YI.,
Sect. Ill., IProp. 1]. Therefore AL, VIR, A'L' are in
GP. Hence AL. A'L' = YR2 = S'Q2 = OQ. KQ; that
is,.pi' A'L' =7r.OQ. Therefore A'L'/-x-= OQ/p,. Hence
Y,(A'L/wr) = OQ Y(1 /p). ]Bnt since A' is the vertex of a
regular polygon whose centre is 0, '~(A'L') = nOQ.
Hence n/w7 =:~(l/p).
Prop. S.-If a transversal through the symmedian point
(K) meet the sides of a harmonic polygon of n sides in the
points R1, Rz,. R,,, and meet the circumcircle in IP;
then I/Ri1P + I/IR2P.... I1/R,,P =n/KP.

flem.-Let a, b, c, &c., be the lengths of the sides;
a, /3, 'y, &c., the perpendiculars on them from the point