Projective geometry, by Oswald Veblen and John Wesley Young.

~~ 18, 19]

QUADRANGULAR SETS

47

configulration is mutual; that is, if either is given, the other is determined.
For a reason which will be evident later, either is called a covariant of the
other.
2. Show that the configuration consisting of two perspective tetrahedra,
their center and plane of perspectivity, and the projectors and traces may be
regarded in six ways as consisting of a complete 5-point P12, P13, 71, P15, P16
and a complete 5-plane 7r456, T2456 7236g, T034, T2345, the notation being
analogous to that used on page 41 for the Desargues configuration. Show
that the vertices of the 5-plane are on the faces of the 5-point.
3. If P1, Po, P3, P4, Pa, are vertices of a complete space 5-point, the ten
points D,, in which an edge pij meets a face PkPIPr (i, j, k, 1, mn all distinct),
are called (iacgonalpoints. The tetrahedra PoPaP4Pr and D Dl3-D14DD5 are perspective with P1 as center. Their plane of perspectivity, r1, is called the polar
of P1 with regard to the four vertices. In like manner, the points P2, P8, P4, P,
determine their polar planes rr2, r3, tr4, 7r. Prove that the 5-point and the polar
5-plane form the configuration of two perspective tetrahedra; that the plane
section of the 5-point by any of the five planes is a quadrangle-quadrilateral
configuration; and that the dual of the above construction applied to the 5-plane
determines the original 5-point.
4. If P is the pole of wr with regard to the tetrahedron A 1A2A3AA4, then is r
the polar of P with regard to the same tetrahedron?
19. The fundamental theorem on quadrangular sets.
THEOREM 3. If two complete quadrangles PPP4IP4 and PI'P'PIP4'
correspond - P to P', P2 to P', etc. - in such a way that five of the
pairs of homologous sides intersect in points of a line 1, then the sixth
pair of homologous sides will intersect in, a point of 1. (A, E)
This theorem holds whether the quadrangles are in the same or
in different planes.
Proof. Suppose, first, that none of the vertices or sides of one of
'the quadrangles coincide with any vertex or side of the other. Let
P P, PP, PP, AP, PP4J be the five sides which, by hypothesis,
meet their homologous sides PJP', P'P', P,'P' P2'P, P'PI in points
of I (fig. 19). We must show that At and P'P' meet in a point
of 1. The triangles PP2P3 and 'P'PI' are, by hypothesis, perspective from 1; as also the triangles i',P4 and P'P'P'. Each pair is
therefore (Theorem 1') perspective from a point, and this point is in
each case the intersection O of the lines PPt' and Pd'. Hence the
triangles P2P and P2''P' are perspective from O and their pairs
of homologous sides intersect in the points of a line, which is evidently 1, since it contains two points of 1. But PPt and ' P ' are