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.

77

Observation.-Triangles whose corresponding vertices lie on
concurrent lines have received different names from geometers.
SALMON and PONCELET call such triangles homologous. These
writers call the point 0 the centre of homology; and the line XYZ
the axis of homology. TOWNSEND and CLEBSCH call them triangles in perspective; and the point 0, and the line XYZ the
centre and the axis of perspective.
Prop. 13.-When three triangles are two by two in
perspective, and have the same axis of perspective, their
three centres of perspective are collinear.
B A C
Let abe, a'b'co, ad' " " be the three As whose corresponding sides are concurrent in the collinear points
A, B, C. Now let us consider the two As aa'a", bb'b",
formed by joining the corresponding vertices a, a', a",
b, b', b", and we see that the lines ab, a'b', a"b" joining
corresponding vertices are concurrent, their centre of
perspective being C. Hence the intersections of their
corresponding sides are collinear; but the intersections
of the corresponding sides of these As are the centres
of perspective of the As abc, a'b'c', a"b"c". Hence the
Proposition is proved.
Cor.-The three As aa'a", bb'b, cc'c" have the same
axis of perspective; and their centres of perspective
are the points A, B, C. Hence the centres of perspective of this triad of As lie on the axis of perspective
of the system abc, a'b'c', a"b"c", and conversely.