# Duality Principle (Projective Geometry)

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## Theorem

### Principle of Duality in the Plane

Let $P$ be a theorem of projective geometry proven using the propositions of incidence.

Let $Q$ be the statement created from $P$ by interchanging:

- $(1): \quad$ the terms
**point**and**(straight) line** - $(2): \quad$ the terms
**collinear**(of points) and**concurrent**(of lines) - $(3): \quad$ the terms
**lie on**and**intersect at**

and so on.

Then $Q$ is also a theorem of projective geometry.

### Principle of Duality in Space

Let $P$ be a theorem of projective geometry proven using the propositions of incidence.

Let $Q$ be the statement created from $P$ by interchanging:

and so on.

Then $Q$ is also a theorem of projective geometry.

## Also known as

A **Duality Principle** is also known as a **Principle of Duality**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**duality**:**1.** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**projective geometry** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**duality**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**projective geometry**