Projective Geometry

Projective geometry studies properties that remain invariant under "projections" (like looking at a drawing from an angle). It simplifies many problems by removing exceptions (like parallel lines never meeting).

The Projective Plane

The Euclidean plane is extended by adding:

Points at Infinity: One for each family of parallel lines. Parallel lines now meet at a point at infinity.
Line at Infinity: The collection of all points at infinity.

Principle of Duality

Theorem (Duality Principle)

Any true statement in projective geometry remains true if we swap:

Points $\leftrightarrow$ Lines
Collinear $\leftrightarrow$ Concurrent
Intersection $\leftrightarrow$ Join

Example: Euclidean:* "Two points determine a unique line." Dual:* "Two lines determine a unique point." (Even parallel lines meet at infinity!).

Desargues' Theorem

Theorem (Desargues' Theorem)

Two triangles

△ A BC

and

△ A^{'} B^{'} C^{'}

are perspective from a point (lines

A A^{'}, B B^{'}, C C^{'}

are concurrent) if and only if they are perspective from a line (intersections of corresponding sides

A B \cap A^{'} B^{'}

BC \cap B^{'} C^{'}

C A \cap C^{'} A^{'}

are collinear).

[Image of Desargues Theorem configuration]

Practice Problems

Exercise (Problem 1)

Use Desargues' Theorem to prove that the three medians of a triangle are concurrent (by considering the triangle and its medial triangle?).

Exercise (Problem 2)

State the dual of the statement: "The three vertices of a triangle lie on a circle."

Exercise (Problem 3)

Explain why a parabola is just an ellipse tangent to the line at infinity.