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.

BOOK VI.

147

the points of contact be G. H, I, J, K, L. Now since
A is the pole of GH, and D A
the pole of JK, the line AD
is the polar of the point of
intersection of the opposite
sides GH and JK of the in- B
scribed hexagon. In like F
manner, BE is the polar of
the point of intersection of L
the lines HI, KL, and CF
the polar of the point of KE J
intersection of IJ and LG;
but the intersections of
the three pairs of opposite
sides of the inscribed hexagon, viz., GH, JK; HI, KL;
IJ, LG, are, by Pascal's Theorem, collinear; therefore
their three polars AD, BE, OF, are concurrent.
(7). If two lines be divided homographically, two lines
joining homologous points can be drawn, each of which
passes through a given point.
For, if AA' (see fig., Prop. 3) pass through a given
point P, join EP, and let fall a I EG on AA'; then
(Cor. 2, Prop. 3) the locus of the point G is a 0; and
since EGP is a right angle, the ( described on EP as
diameter passes through G; hence G is the point of
intersection of two given Os; and since two Os intersect in two points, we see that two lines joining homographic points can be formed, each passing through P.
Now, if we reciprocate the whole diagram with respect
to a circle whose centre is P, the reciprocals of the
points A, A' will be parallel lines. Hence we have the
following theorem in a system of two homographic
pencils of rays:-There exist two pairs of homologous
rays which are parallel to each other.
Cor.-There are two directions in which transver,
sals can be drawn, intersecting two homographic pencils of rays so as to be divided proportionally, namely,
parallel to the pairs of homologous rays which are
parallel.