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.

38

A SEQUEL TO EUCLID.

AOB, BOO, COA = four right Zs;.. the Z COA
= AFB + BDC: to each D
add the Z CEA, and we
have the Z COA + CEA
= sum of the three Ls of
the A DEF, that is = two
right Zs;.'. the quadrilateral AECO is in-
scribed in a (. Hence the
three Os pass through
a common point, which
is a given point.
Again, since the area \ -7
of the A DEF is a maximum, each of its sides is a maximum. Hence (15) we
have to draw through the point A a line II to the line
joining the centres of the Os ABF, CEA; that is, a
line J_ to AO, and join its extremities E, F to the
points C, B, respectively.
Cor.-If instead of the maximum A we require to
describe a A whose sides will be equal to three given
lines, the method of solving the question can be inferred
from the corollary to the last Proposition.
Prop. 17.-To describe in a given triangle DEF (see
last fig.) a triangle given in species whose area shall be a
minimum.
Analysis.-LetABC be the inscribed A; describe 0(s
about the three As ABF, BCD, CAE; then these Os
will have a common point: let it be O. We prove this
to be a given point as follows: The Z FOE exceeds
the Z FDE by the sum of the Zs DFO, DEO; that is,
by the sum of the Zs BAO, CAO. Hence the Z FOE
= FDE + BAC;.-. the Z FOE is given. In like
manner, the Z EOD is given. Hence the point O will
be the point of intersection of two given Os, and is.'. given; and, since E and F are given points, the
Z OFE is given;.'. the Z OBA is given. In like
manner, the Z OAB is given;.'. A OAB is given