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.

170

170 ~A SEQUEL TO EUCLID.

For if LMN be the triangle formed by the tangents,
the angle M1AC is [-Euc. III. xxxii.] equal to ABC;
therefore MAC = AC'B', and hence AMi is parallel to
C'B'.
Cop.. 1-[The _points B', C'; B, C are concyclio.
CoR. 2.-The lines joining the feet of the, perpendicula~rs of a triangle are antiparallel to its sides.
IDEr.-The isogonal corjutgate of a median AM of a
triangle is called a symmedian.
It follows from Prop. 2 that the three symmedians
of a triangle are concurrent. The point of concurrenace (1K) is called the symmedian point of the, triangle.
Prop. 5.-Th~e perpendiculars from K on the sides of
the triangle are pro~portional to the sides.
Dem.-Let the perpendiculars from M on the sides
AB, AC be MID) ME, and from K on the three sides
x, y, z. Then [Prop. 1, 10] MID. z= ME. Y but MID.
AB = ME. AC. Hence y:z AC: AB, which proves
the proposition.