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.

146

A SEQUEL TO EUCLID.

(3). Lines drawn from any variable point in the plane
of a quadrilateral to the six points of intersection of its
four sides form a pencil in involution.
This Proposition is evidently the reciprocal of (1),
Prop. 9, Section VI. The F
following is a direct proof:
Let ABCD be the quadrilateral, and let its opposite /> c
sides meet in the points E /
and F, and let 0 be the point B
in the plane of the quadri- \
lateral; the pencilfromOto E
the points A, B, C, D, E, F 0 y
is in involution.
Dem.-Join OE, cutting the sides AD, BC in X and
Y. Join EF. Now, the pencil(O.XADF)=(E. XADF)
= (E. YBCF)=(O. YBCF)=(O. XBCF);.-. (O. EADF)
= (0. EBCF). Hence the pencil is in involution.
(4). If two vertices of a triangle move on fixed lines,
while the three sides pass through three collinear points,
the locus of the third vertex is a right line. Hence, reciprocally, If two sides of a triangle pass through fixed
points, while the vertices move on three concurrent lines,
the third side will pass through afixed point.
(5). To describe a triangle about a circle, so that its three
vertices may be on three given lines. This is solved by
inscribing in the circle a triangle whose three sides
shall pass through the poles of the three given lines,
and drawing tangents at the angular points of the inscribed triangle.
(4). BRIANCHON's THEOREM.-If a hexagon be described
about a circle, the three lines joining the opposite angular
points are concurrent.
This is the reciprocal of Pascal's Theorem: we prove
it as follows:Let ABCDEF be the circumscribed hexagon; the
three diagonals AD, BE, CF are concurrent, For, let