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
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
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
Given: point P not on line k.
Where is the flaw?
Let Q be the foot of the perpendicular from P to k.
Let m be the line through P perpendicular to line PQ
Line m is parallel to line k.
Let n be any line through P distinct from m and line
Let ray PR be a ray of n between ray PQ and a ray
of m emanating from P.
There is a point A between P and Q.
Let B be the unique point such that Q is between A
and B and AQ is congruent to QB.
Let S be the foot of the perpendicular from A to n.
Let C be the unique point such that S is between A and C
and AS is congruent to SC.
A, B, and C are not collinear.
There is a unique circle G passing through A, B, and
k is the perpendicular bisector of AB, and n is the
perpendicular bisector of AC.
k and n meet at the center of G.
The parallel postulate has been proven.
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
After observing the construction, manipulate
the free (red-colored) points to find the flaw:
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