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.

48

A SEQUEL TO EUCLID.

Analysis.-Let ABC be the A, CD the line bisecting the vertical Z. Then AD, DB,
and the Z ACB are given. Now,
since AD, DB are given, AB is
given; and since AB and the Z ACB
are given, the ( ACB is given A D
(xxxiii.); and since CD bisects the
L ACB, we have arc AE = EB;.'. E is a given point, and D is a E
given point. Hence the line ED is given in position,
and therefore the point C is given.
(3). Given the base, the vertical angle, and the rectangle
of the sides, construct the triangle.
Analysis.-Let ABC be the A;
let fall the 1 CD; draw the diameter CE; join AE. Now the
Z CEA = CBA (xxi.), and CAE
is right, being in a semicircle
(xxxi.);. Z. =L CDB. Hence the A ----
As CAE, CDB are equiangular;.-. rectangle AC. CB = rectangle
CE. CD (9); but rectangle AC. CB E
is given;.'. rectangle CE. CD is given; and since the
base and vertical Z are given, the G ACB is given;.. the diameter CE is given;.. CD is given; and therefore the line drawn through C 11 to AB is given in
position. Hence the point C is given.
The method of construction is obvious.