4.6 Investigation into the Uniqueness of a Projectivity between Pencils of Points
Teach to the problems, not to the text.
—Kim Nebeuts, Return to Mathematical Circles (1988)

Theorem 4.10. If A, B, C and A', B', C' are distinct elements in pencils of points with distinct axes p and p', respectively, then there exists a projectivity such that ABC is projectively related to A'B'C'.

The theorem and its constructive proof, gave a procedure to determine a corresponding point D' on axis p' by following the perspectivities when a fourth point D on axis p was given. That is, let D be an element of axis p, then let D₁ = DP · A'C and D' = D₁ Q · p'. Is the point D' unique? Or, does the point D' depend on the choice of the point P?

Drag the red point P to investigate further.
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