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.

148

A SEQUEL TO EUCLID.

(8). If we reciprocate Prop. 3 we have the following
theorem:-Being given afixed point, namely, the centre
of the circle of reciprocation and two homographic pencils
of rays, two lines can be found (the polars of the points E
and F in Prop. 3), so that the portions intercepted on
each by homologous rays of the pencils will subtend an angle
of constant magnitude at the given point.
SECTION VIII.
MISSCELLANEOUS EXERCISES.
1. The lines from the angles of a A to the points of contact of
any 0 touching the three sides are concurrent.
2. Three lines being given in position, to find a point in one of
them, such that the sum of two lines drawn from it, making given
angles with the other two, may be given.
3. From a given point in the diameter of a semicircle produced
to draw a line cutting the semicircle, so that the lines may have a
given ratio which join the points of intersection to the extremities
of the diameter.
4. The internal and external bisectors of the vertical angle of
a A meet the base in points which are harmonic conjugates to the
extremities.
5. The rectangle contained by the sides of a A is greater than
the square of the internal bisector of the vertical angle by the
rectangle contained by the segments of the base.
6. State the corresponding theorem for the external bisector.
7. Given the base and the vertical angle of a A, find the folowing loci:(1). Of the intersection of perpendiculars.
(2). Of the centre of any circle touching the three sides.
(3). Of the intersection of bisectors of sides.
8. If a variable 0 touch two fixed Os, the tangents drawn to it
from the limiting points have a constant ratio.