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.

16

A SEQUEL TO EUCLID.

Cor. 3.-Of all polygons whose vertices lie on fixed
lines, that of minimum perimeter is the one whose
several angles are bisected externally by the lines on
which they move.
Prop. 22.-Of all triangles having the same base and
area, the perimeter of an isosceles triangle is a minimum.
Dem.-Since the A s are all equal in area, the vertices
must lie on a line II to the base, and the sides of an
isosceles A will evidently make equal Zs with this
parallel; hence their sum is a minimum.
Cor.-Of all polygons having the same number of
sides and equal areas, the perimeter of an equilateral
polygon is a minimum.
Prop. 23.-A large number of deducibes may be given
in connexion with Euclid, fig., Prop. xlvii. Me insert a
few here, confining ourselves to those that may be proved
by the First Book.
(1). The transverse lines AE, BK are J to each
other. For, in the As ACE, BCK, which are in every
respect equal, the Z EAC o
= BKC, and the Z AQO
= KQC; hence the angle
AOQ = KCQ, and is.'. a \/
right angle.\\ \
(2). AKCE = DBF.
Demn. — Produce KC,
and let fall the ~ EN. \ B
Now, the Z ACN = BCE,
each being a right angle;.. the ZACB = ECN, and
ZBAC=ENC,eachbeing
a right angle, and side
BC = CE; hence (xxvi.)
EN = AB and CN = A; but AC = CK;.*. CN = CK, and
the A ENC = ECK (xxxviii.); but the A ENC = ABC;
hence the A ECK = ABC. In like manner, the
I DBF = ABC;.-. the A ECK = DBF.
(3). EK2 + FD2 = 5BC2.
Dem.-EK2 = EN2 + NK2 (xlvii.);