Exercise 4.38. Investigation for an Alternate Construction of a Projectivity  Acrobat Reader IconPrintout
Art, like morality, consists of drawing the line somewhere.
Exit book to another website.G. 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 GeoGebra or JavaSketchpad. Revise your conjecture, if needed.
You may also explore using the prepared Geometer's Sketchpad sketch found in the Appendix - Sketchpad Sketches.

4.6.3 Harmonic Sets and ProjectivityBack to Harmonic Sets and ProjectivityNext to Alternate Construction of a Projectivity4.6.4 Alternate Construction of a Projectivity

Ch. 4 Projective TOC  Table of Contents

  Timothy Peil  Mathematics Dept.  MSU Moorhead

© Copyright 2005, 2006 - Timothy Peil