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.

78

A SEQUEL TO EUCLID.

Prop. 14. —When three triangles which are two by two
in perspective have the same centre of homology, their three
axes of homology are concurrent.

Let abc, a'b'c', a"b"c" be three As, having the point
0 as a common centre of perspective. Now, let us
consider the two A s formed by the two systems of
lines ab, a'b, a"b"; and ac, a'o, a"o"; these two As are
in perspective, the line Oaa'a" being their axis of
perspective. Hence the line joining their corresponding vertices are concurrent, which proves the Proposition.
Cor.-The two systems of As, viz., that formed by
the lines ab,, a"b"; be, bic', b"c"; ca, c'a', o'a"; and
the system abc, a'b'c', a "bc", have corresponding properties-namely, the three axes of perspective of either
system meet in the centre of perspective of the other
system.
Prop. 15.-We shall conclude this section with the
solution of a few Problems: