Exercise 4.38. Investigation for an Alternate Construction of a Projectivity
Art, like morality, consists of drawing the line somewhere.
—C. K. Chesterton (1874–1936)

        The construction of the proof of Theorem 4.10 requires several steps to construct a projectivity between two pencils of three points. Another method is often used that is more efficient. In the given diagram, the two pencils of points are projectively related, . Draw the lines and consider the points AB' · BA', AC' · CA', AD' · DA', BC' · CB', BD' · DB', and CD' · DC'. (The pairs of lines, you drew, are called cross joins.)

        (a) How are the points related to each other?

        (b) State your result as a conjecture.

        (c) Now let E be an arbitrary point on the pencil of points with A, B, C, D. Based on your conjecture, construct the image of E.

Explore further with a dynamic interactive diagram. Revise your conjecture, if needed.
You may also explore using the prepared Geometer's Sketchpad sketch found in the Appendix - Sketchpad Sketches.