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.

73

Let ABC be the A which is given in species; let
the point A be fixed; the point B move along a given
line BD: it is required to find the
locus of C. C
From A let fall the -L AD on
BD; on AD describe a A ADE
equiangular to the A ABC; then B
the L ADE is given in position;.'. E is a given point. Join EC.
Now, since the As ADE, ABC D
are equiangular, we have
AD: AE:: AB: AC;
therefore AD:AB:: AE: AC:
and the angle DAB is evidently = EAC. Hence the A s
DAB, EAC are equiangular;.-. the angle ADB = AEC.
Hence the angle AEC is right, and the line EC is given
in position;.-. the locus of C is a right line.
Cor. —By an obvious modification of the foregoing
demonstration we can prove the following theorem:If a A be given in species, and have one angular point
given in position; then if a second angular point move
along a given 0, the locus of the third angular point
is a circle.
Prop. 10. —If 0 be the centre of the inscribed circle of
the triangle ABC, then AO2: AB. AC:: s - a: a.
Dem.-Let O'be the centre A
of the escribed 0 touching
BC externally; let fall the/
I-s OD, O'E. Join OB, OC,
O'B, O'C. Now, the Zs /
O'BO, O'CO are evidently
right Zs;.-. OBO'C is a
quadrilateral inscribed in a
circle, and Z BO'O = BCO
= ACO; and BAO' = OAC.
Hence the triangles O'BA and
COAare equiangular;.'. O'A:
BA:: AC: AO;.*. O'A OA = AB AC. A H ence