Math 487
Chapter 4 Practice Problems
1. Give the definition of each of the following terms:
(a) A complete quadrangle
(b) A complete quadrilateral
(c) A perspectivity between pencils of points
(d) A perspectivity between pencils of lines
(e) A projectivity between pencils of points
(f) The harmonic conjugate of a point Cwith respect to points Aand B.
(g) A point conic
(h) A line conic
2. State each of the following:
(a) Desargues’ Theorem
(b) The Fundamental Theorem of Projective Geometry
3. True or False
(a) In a plane projective geometry, if two triangles are perspective from a point, then they are also perspective from
a line.
(b) In the Poincar´e Half Plane, if two triangles are perspective from a point, then they are also perspective from a
line.
(c) In a plane projective geometry, if two triangles are perspective from a line, then they are also perspective from a
point.
(d) Every point in a plane projective geometry is incident with at least 4 distinct lines.
(e) If H(AB, C D) then H(CD, B A).
(f) If H(AB , CD) and H(AB , C′D) then C=C′
(g) If A, B, C and A′, B′, C′are distinct elements in pencils of points with distinct axes pand p′, there there exists a
perspectivity such that ABC
o
−
ˆA′B′C′
4. Prove that Axiom 3 in independent of Axiom 1 and Axiom 2.
5. (a) State and prove the dual of Axiom 3.
(b) State and prove the dual of Axiom 4.
6. (a) Prove that a complete quadrangle exists.
(b) Draw a model for a complete quadrangle EF GH .
(c) Identify the pairs of opposite sides in the quadrangle EF GH .
(d) Construct and identify the diagonal points of the quadrangle EF GH .
7. (a) Prove that a complete quadrilateral exists.
(b) Draw a model for a complete quadrilateral abcd.
(c) Identify the pairs of opposite points in the quadrilateral abcd.
(d) Construct and identify the diagonal lines of the quadrilateral abcd.
8. (a) Construct an example of two triangles that are perspective from a point. Be sure to identify the point Othat the
triangles are perspective from.
(b) Are these two triangles also perspective from a line? If so, identify the line that the triangles are perspective from.
If not, explain why they cannot be perspective from a line.
9. Illustrate a projectivity from a pencil of lines a, b, c with center Oto a pencil of lines a′, b′, c′with center O′6=O.
