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 V1.

71

the six lines OA, OB... OF be denoted by I, m, n, p,
g, r; then we have Da = Im; Dy = np; De = yr;
therefore D3aye = lmnpqr.
In like manner, D3ps,8 = lmnpqr;
therefore aye = /80. (Q.E.D.)
Cor. 1.-The six points A, B, C, D, E, F may be
taken in any order of sequence, and the Proposition
will hold; or, in other words, if we draw all the
diagonals of the hexagon, and take any three lines,
such as AC, BD, EF, which terminate in the six points
A, B, C, D, E, F, then the continuous product of the
I-s on them will be equal to the continuous product of
the I-s on any other three lines also terminating in the
six points.
Cor. 2.-When the figure inscribed in the circle contains only four sides, this Proposition is the theorem
proved (III., 11.)
Cor. 3.-If we suppose two of the angular points to
become infinitely near; then the line joining these
points, if produced, will become a tangent to the circle,
and we shall in this way have a theorem that will be
true for a polygon of an odd number of sides.
Cor. 4.-If perpendiculars be let fall from any point
in the circumference of a circle on the sides of an inscribed triangle, their continued product is equal to the
continued product of the perpendiculars from the same
point on the tangents to the circle at the angular points.
Prop. 7.-Given, in magnitude and position, the base
BC of a triangle and the
ratio BA: AC of the
sides, it is required to
find the locus of its A
vertex A.
Bisect the internal B
and the external verti-\
cal angles by the lines
AD, AE. Now, BA:
AC:: BD:DC (III.);
but the ratio BA: AC is given (Hyp.); therefore the