Projective geometry, by Oswald Veblen and John Wesley Young.

54

PROJECTION, SECTION, PERSPECTIVITY [CHAP. II

17.. A plane section of a 6-point in space gives (in six ways) a 5-point whose
10 3.
sides pass through the points of a configuration
3 10
18. A plane section of an n-point in space gives a complete (n - 1)-point
whose sides pass through the points of a configuration n-1C2 a - 3
3 n-1C3
* 19. The n-space section of an m-point (in - n + 2) in an (n + l)-space can be
considered in the n-space as (m- k) — points (in mC,_. ways) perspective in pairs
from the vertices of the n-space section of one (n - k)-point; the r-spaces of
the k-point figures meet in (r - 1)-spaces (r = 1, 2, *, n - 1) which form, the
n-space section of a k-point.
* 20. The figure of two perspective (n + 1)-points in an n-space separates
(in n + 3 ways) into two dual figures, respectively an (n + 2)-point circuniscribing the figure of (n + 2) (n- 1)-spaces.
21. The section by a 3-space of an n-point in 4-space is a configuration,C2 n-2 -,,C2
3 nC3 n-3
6 4,n4
The plane section of this configuration is
nC3 n-3
4 nC,
22. Let there be three points on each of two concurrent lines 11, 1,. The
nine lines joining points of one set of three to points of the other determine
six triangles whose vertices are not on 11 or 12. The point of intersection of 11
and 12 has the same polar with regard to all six of these triangles.
23. If two triangles are perspective, then are perspective also the two
triangles whose vertices are points of intersection of each side of the given
triangles with a line joining a fixed point of the axis of perspectivity to the
opposite vertex.
24. Show that the configuration of the two perspective tetrahedra of
Theorem 2 can be obtained as the section by a 3-space of a complete 6-point
in a 4-space.
* 25. If two 5-points in a 4-space are perspective from a point, the corresponding edges meet in the vertices, the corresponding plane faces meet in the
lines, and the corresponding 3-space faces in the planes of a complete 5-plane
in a 3-space.
* 26. If two (n + 1)-points in an n-space are perspective from a point,
their corresponding r-spaces meet in (r - 1)-spaces which lie in the same
(n - 1)-space (r = 1, 2 *, n - 1) and form a complete configuration of
(n + 1) (n - 2)-spaces in (n - 1)-space.