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.

64 A SEQUEL TO EUCLID.
that is, 2sr =2ALABC;
therefore 8r= A~ ABC. (a)
Cor. 2.-If the 0D touch the side BC externally, and
the sides AB, AC produced; thatA
is, if it be an escribed 0), and ifA
the points of contact be denoted
by D', E', F', it may be proved
in the same manner that AE'o
-AF'-=8; BD'.=BF' = 8 -C C D
These Propositions, though simple, are very important. P
(Cor. 3.-If r' denote the radius0
of the escribed 0), which touches
the side BC (a) externally,
r'(s - a) = A ABC.
Demn.- E'O'. AC = 2AAO'C;
that is r'.b= 2,AAO'C.
In like manner r' c = 2 A AO'B,
and r.a = 2ALBO'C.
Hence r'(b +e- a) = 2 ABC;
that is r'.2 ( - a) = 2AABC;
therefore r'. (8 - a) = A ABC. (3
C'or. 4.-The rectangle r. r' = (8 - b) (8 - o).
Dem.-Since CO bisects the L ACB, and C0' bisects
the L BCE', CO is at right L s to 00';.-. the Z EGO
+ E'CO' = a right L_; and L ECO + COE = one right
L_;.-. E'CO' = COE. Hence the As E'CO', EOC are
equiangular;- and, therefore,
E'O. EO = E'C. CE; (III. 9.)
that is r. r' = (8 - b) (8 - o). (Y)