The Hyperbolic Toolbox:
Non-Euclidean Constructions in Geometer's Sketchpad

Example 6.2:  "Proving the Parallel Postulate"

The geometry that essentially arises from Euclid's first four postulates (see the Introduction) is called Neutral (or Absolute) Geometry.  Historically, before the discovery of hyperbolic geometry,  there were  numerous attempts to prove Euclid's fifth postulate in neutral geometry.  Of course, these attempts were doomed to failure since hyperbolic geometry provides an example of  a neutral geometry in which Euclid's fifth postulate is violated (noting that hyperbolic geometry is no less consistent than Euclidean geometry).

One type of activity that can help students understand the axiomatic distinction between hyperbolic and Euclidean geometry is considering historical attempts to prove Euclid's fifth postulate in neutral geometry.  Each of these flawed attempts contains at least one unjustifiable (in neutral geometry) statement, else the proof would be valid and Euclid's fifth postulate would be a theorem.  The flawed statement logically implies and most often is equivalent to the parallel postulate.  A terrific exercise for students is to consider one of these proofs, with the justifications left out.

Below is an attempted proof of the parallel postulate in neutral geometry. This "proof" was the work of Farkas Bolyai, who was the father of Janos Bolyai, one of the discoverers of hyperbolic geometry.  Your job is to justify all the steps that can be justified in neutral geometry and to identify the statement that is equivalent to the parallel postulate (i.e. find the flaw!). Be complete! Some statements might require more than one justification.

Given:  point P not on line k.

      1. Let Q be the foot of the perpendicular from P to k.
      2. Let m be the line through P perpendicular to line PQ
      3. Line m  is parallel to line k.
      4. Let n be any line through P distinct from m and line PQ.
      5. Let ray PR be a ray of n between ray PQ and a ray of m emanating from P.
      6. There is a point A between P and Q.
      7. Let B be the unique point such that Q is between A and B and AQ is congruent to QB.
      8. Let S be the foot of the perpendicular from A to n.
      9. Let C be the unique point such that S is between A and C and AS is congruent to SC.
      10. A, B, and C are not collinear.
      11. There is a unique circle G passing through A, B, and C.
      12. k is the perpendicular bisector of AB, and n is the perpendicular bisector of AC.
      13. k and n meet at the center of G.
      14. The parallel postulate has been proven.

      Where is the flaw?

    How the tools can help:

      Students with some experience in geometry from an axiomatic standpoint will typically be able to recognize many of  the above statements as propositions in neutral geometry and justify those statements.  However, they may have difficulty precisely identifying the unjustifiable statement and how it is flawed.  When students are stuck, it helps to encourage them to try reproducing the proof in one of their hyperbolic models.  The statement that is unjustifiable in neutral Geometry becomes false in hyperbolic geometry.

      Below is a Java Sketchpad script that illustrates the proof's constructions in the Klein model.  (This particular applet may take a few moments to load.)  Click on "Construction Steps"  then click on each step in turn to see the construction. After observing the construction, manipulate the free (red-colored) points to find the flaw:

The dynamic figures on this page were produced using JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright ©1990-1998 by Key Curriculum Press, Inc. All rights reserved. 

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