Solution Exercise 4.31(b).
The most incomprehensible fact about the universe is that it is comprehensible.
—Albert Einstein (1879–1955)

(b)  Assume a, b, c are elements of a pencil with center P and a', b', c' are elements of a pencil with center P'. Use dynamic geometry software to construct the projectivity such that abc is projectively related to a'b'c'.

Type the action buttons in order to illustrate the steps of the constructive proof of the dual of Theorem 4.10.
To reset to the original settings, type the letter "R" on the keyboard.

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To investigate further, drag points in red that define the pencils of lines to various locations.
To investigate the projectivity, drag point d the point defining an arbitrary line in the pencil with center P. Note what happens to the image line d' when d is dragged to lines a, b, and c.
To reset to the original settings, type the letter "R" on the keyboard.