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.

208

A SEQUEL TO EUtLID.

Dem.-Let AB, A'B', produced, intersect in P,
and meet the line OK in C, C'. Now the polar of S
will pass through P and through S'; then the pencil
P(SCS'C') is harmonic..2. 2/SS' = 1/SC +/SC SC' = 1/SK 1/SO.
Hence (SK - SC) / (SK. SC) = (SC' - SO) / (SC'. SO);.. KC/SC: OC'/SC::: SO;
or, (K, AB) / (S, AB): (0, A'B') /(S, A'B'):: SK: SO.
But (S, AB): (S, A'B'):: AB: A'B'. [Book VI.,
Sect. iv., Prop. 6.]
Hence (K, AB) / AB: (0, A'B') / A'B':: SK: SO.
Now, since AB is a side of a harmonic polygon
whose symmedian point is K, the ratio (K, AB) / AB is
constant; and since S, K, O are given points, the ratio
SK: SO is given; hence the ratio (0, A'B') / A'B' is
constant;.'. A'B is constant. Hence the proposition
is proved.
Cor. 1.-If we join the points A, B to S', and produce
to meet Z again in A", B", the points A", B" are the
reflexions of A, B, with respect to the diameter DD' of
Z.
DEF. vii.-The points S, S' are called the centres of
inversion of the harmonic polygon.
COR. 2,- The centres of inversion of a harmonic polygon
are harmonic conjugates with respect to its circumcentre
and symmedian point.
Observation.-It is evident that this proposition
gives a new demonstration of Prop. i. It is also plain,
if, instead of 0, we take a point K' collinear with
O and K, and repeat the foregoing construction, only
using K' instead of O, that we shall have the harmonic
polygon, whose symmedian point is K, inverted into
another whose symmedian point is K'.