Projective geometry, by Oswald Veblen and John Wesley Young.

~ 36]

CONFIGURATION OF PAPPUS

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It is known as the configuration of Pappus.* It should also be noted
that this configuration lnay be considered as a simple plane hexagon
(six-point) inscribed in two intersecting lines. If the sides of such a
hexagon be denoted in order by 1, 2, 3, 4, 5, 6, and if we call the sides
1 and 4 opposite, likewise the sides 2 and 5, and the sides 3 and 6 (cf.
Chap. II, ~ 14), the last theorem may be stated in the following form:
COROLLARY. If a simple hexagon be inscribed in two intersecting lines,
the three pairs of opposite sides will intersect in collinear points.t
Finally, we may note that the nine points of the configuration of
Pappus may be arranged in sets of three, the sets forming three
triangles, 1, 2, 3, such
that 2 is inscribed in
1, 3 in 2, and 1 in 3.
This observation leads
to another theorem connected with the Pappus
configuration.
THEOREM 22. If
A,,B2C, be a triangle A1 C B
ilscribed in a triangle FIG. 45
A1B1C1, there exists a
certain set of triangles each of which is inscribed in the former and
circumscribed about the latter. (A, E, P)
Proof. Let [a] be the pencil of lines with center A1; [b] the pencil
with center B,; and [c] the pencil with center C, (fig. 45). Consider the
BA2 B C
perspectivities [a] [b] b 2 [c]. In the projectivity thus established between [a] and [c] the line A1C1 is self-corresponding; the
pencils of lines [a], [c] are therefore perspective (Theorem 17, Cor.
(dual)). Moreover, the axis of this perspectivity is C2AQ; for the lines
A1C2 and C1C2 are clearly homologous, as also the lines A1A2 and C1A2.
Any three homologous lines of the perspective pencils [a], [b], [c] then
form a triangle which is circumscribed about A1B1C1 and inscribed
in AOBC.
* Pappus, of Alexandria, lived about 340 A.D. A special case of this theorem may
be proved without the use of the fundamental theorem (cf. Ex. 3, p. 52).
t In this form it is a special case of Pascal's theorem on conic sections
(cf. Theorem 3, Chap. V).