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.

72

A SEQUEL TO EUCLID.

ratio BD: DC is given, and BC is given (Hyp.);.'. the
point D is given. In like manner the point E is given.
Again, the angle DAE is evidently equal half the sum
of the angles BAG, CAP. Hence DAE is right, and the
circle described on the line DE as diameter will pass
through A, and will be the required locus.
Cor. 1.-The circle described about the triangle ABC
will cut the circle DAE orthogonally.
For, let O be the centre of the 0 DAE. Join AO;
then the angle DAO = ADO, that is, DAC + CAO
= BAD + ABO; but BAD = DAC;.~. CAO = ABO;.-. AO touches the 0 described about the L BAC.
Hence the Os cut orthogonally.
Cor. 2.-Any circle passing through the points B, C,
is cut orthogonally by the circle DAE.
Cor. 3.-If we consider each side of the triangle as
base in succession, the three circles which are the loci
of the vertices have two points common.
Prop. 8.-If through 0, the intersection of the diagonals of a quadrilateral ABCD, E
a line OH be drawn parallel
to one of the sides AB, meeting
the opposite side CD in G, and D
the third diagonal in H, OH
is bisected in G. CI \
Dem. —Produce HO to /
meet AD in I, and let it
meet BC in J A B
Now IJ: JH:: AB: BF, (Prop. 3.)
and OJ: JG:: AB: BF;
therefore IO: G:: AB: BF;
but AB: BE:: IO: OG;. OG= GH.
Cor.-GO is a mean proportional between GJ and
GI.
Prop. 9. —If a triangle given in species have one angular point fixed, and if a second angular point moves along
a given line, the third will also move along a given line.