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.

6

A SEQUEL TO EUCLID.

A and B be two fixed points, and if a variable point P
moves so that the area of the A ABP retains the same
value during the motion, the locus of P will be a right
line 1I to AB.
Prop. 8. —IfAB, CD be two lines given in position
and magnitude, and if a point P moves so that the sum
of the areas of the triangles ABP, CDP is given, the locus
of P is a right line.
Dem.-Let AB, CD intersect in 0; then cut
off OE = AB, and OF = CD; join OP, EP, EF,
FP; then A APB D
= OPE, and CPD =
OPF; hence the sum
of the areas of the As
OEP, OFP is given; /.*. the area of the F
quadrilateral OEPF is
given; but the A OEF
is evidently given;..
the area of the A EFP 0 A SG B
is given, and the base V".
EF is given;.. the locus of P is a right line II to EF.
Let the locus in this question be the dotted line in the diagram.
It is evident, when the point P coincides with R, the area of the
A CDP vanishes; and when the point P passes to the other side
of CD, such as to the point T, the area of the A CDP must be
regarded as negative. Similar remarks hold for the A APB and
the line AB. This is an instance of the principle (see 5, note)
that the area of a A passes from positive to negative as compared
with any given A in its own plane, when (in the course of any
continuous change) its vertex crosses its base.
Cor. 1.-If m and n be any two multiples, and if
we make OE = mAB and OF = nCD, we shall in a
similar way have the locus of the point P when m times
A ABP + n times CDP is given; viz., it will be a right
line 11 to EF.
Cor. 2.-If the line CD be produced through 0, and
if we take in the line produced, OF' = nCD, we shall
get the locus of P when m times L ABP - n times
CDP is given.
Cor, 3.-If three lines, or in general any number of