** 4.3 Duality in Projective Geometry
***Printout*

**I think and think for months and years,
ninety-nine times, the conclusion is false. The hundredth time I am right.**

**
—**

Remember that the dual of a statement was defined in the section on axiomatic systems in the first chapter.

In this section, we show that plane projective geometry satisfies the principle of duality. Remember that once we have proven a theorem, by the principal of duality, the dual of the theorem is also valid, i.e., one proof proves two statements.

Note that the first two axioms are nearly the duals of
each other.

** Axiom 1**. Any two distinct points are incident with exactly one line.

** Axiom 2**. Any two distinct lines are incident with at least one point.

In the last section on basic theorems, the dual of
Axiom 1 was stated and you worked out its
proof.

**Dual of Axiom 1.** **Any two distinct lines are incident with exactly one
point.**

* Exercise 4.11.* For Axiom 2,
write the dual and its proof..

Note that neither of the first two axioms or their duals provide for the existence of any points or lines. However, Axiom 3 and its dual state that at least four points and four lines exist.

**Dual of Axiom 3**. **There exist at least four lines, no three of which are
concurrent.**

*Proof. *Let *A, B, C, *and *D* be four distinct points, no
three collinear; the existence of these points is given by
Axiom 3. Thus by
Axiom 1, and since no three of the points are collinear, there are six distinct
lines* AB, AC, AD, BC, BD,* and * CD.*

Consider the four lines* AB, BC,
CD,* and* DA.* We assert that no three of these lines are concurrent.
Suppose not, then three of the lines would be concurrent, say *AB, BC, *and
*CD* are concurrent. By the Dual of Axiom 1, *B* is the only
point of intersection of *AB* and* **BC*. Hence *B*
is the point of concurrency for the three lines *AB, BC, *and *CD*.
Thus *B* is on *CD, *which contradicts the initial assumption that *
B, C, *and *D* are noncollinear. The other cases follow from a similar
argument.

Therefore, there exist at least four
lines, no three of which are concurrent. //

Before examining the Dual of Axiom 4, we need to define a complete quadrilateral, which is the dual of a complete quadrangle. Also, note some of the differences between complete quadrangles and complete quadrilaterals as well as differences between complete quadrilaterals and Euclidean quadrilaterals.

** Definition.** A

*
Model.* The sides of the quadrilateral* abcd *are *
a, b, c *and *d.*
The vertices of the quadrilateral are *E = a ^{ }
· b, F *=

Note that unlike a triangle, which is similar in definition to a Euclidean triangle, the quadrangle and quadrilateral do not have similar analogues in Euclidean geometry. Also, unlike in Euclidean geometry, the quadrangle and quadrilateral are different figures.

**Dual of Axiom 4**. **The three diagonal lines of a complete quadrilateral are never
concurrent.**

*Proof. *Let *abcd* be a complete quadrilateral; the existence is
given by the Dual of Axiom 3. Let *E = a · ^{ }b, F *=

Click here for a dynamic illustration of Desargues' Theorem with either GeoGebra or JavaSketchpad.

* Dual of Axiom 5. (Dual of Desargues' Theorem) If two triangles are perspective
from a line, then they are perspective from a point.*
Assume triangle

Proof.

Consider triangles

Definitions for perspectivity and projectivity for pencils of points and lines.

**Dual of Axiom 6**. **If a projectivity on a pencil of lines leaves three distinct
lines of the
pencil invariant, it leaves every line of the pencil invariant.**

**Exercise 4.12**.

(a) How many cases are there in the
proof of the Dual of Axiom 3?

(b) State the other cases.

(c) Prove at least one of the cases.

* Exercise 4.13.
*(a) Prove the existence of a complete quadrilateral.

(b) What are similarities and differences between complete quadrilaterals and complete quadrangles.

(c) What are similarities and differences between complete quadrilaterals and Euclidean quadrilaterals.

* Exercise 4.14.
*(a)

(b) Prove that every point is incident with at least four distinct lines.

* Exercise 4.15. *
Prove the Dual of Axiom 6.

© Copyright 2005, 2006 - Timothy Peil |