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 VI.

79

(1). To describe a rectangle of given area, whose four
sides shall pass through four given points.
Analysis.-Let ABCD be A G I B
the required rectangle; E, F,.
G, H the four given points. J
Through E draw EI 1I to AD; \ 0
and through H draw HJ 11 to \
AB, and HO L to EG; and X F c
draw JK ~L to HO produced.
Now it is evident that the As EIG, JHK, are equiangular;.-. the rectangle El. JH = EG. HK; but
EI. JH = area of rectangle, and is given;.'. the rectangle EG. HK is given, and EG is given;.'. HK is
given. Hence the line KJ is given in position; and
since the angle FJH is right, the semicircle described
on HF will pass through J, and is given in position.
Hence the point J, being the intersection of a given
line and a given 0, is given in position; therefore the
line FJ is given in position.
(2). Given the base of a triangle, the perpendicular, and
the sum of the sides, to construct it.
Analysis.-Let ABC be the A, CP the 1.; and let
DE be the diameter of the circumscribed 0, which is - to E
AB; draw CH II to AB.
Now the rectangle DH. EG is
equal to the square of half sum
of the sides (IV., 8);.'. DH.EG F
is given; and DG. GE = square A
of GB, and is given. Hence G /p B
the ratio of DH. GE: DG. GE
is given;.'. the ratio of DH D: DG is given. Hence the ratio D
of GH: DG is given; but GH is = to the L, and is
given; hence DG is given; then, if AB be given in
position, the point D is given;.'. the ( ADB is given
in position, and CE at a given distance from AB is
given in position. Hence the point C is given in position.