The Hyperbolic Toolbox:
Non-Euclidean Constructions in Geometer's Sketchpad
Example 6.2: "Proving the Parallel Postulate" (Solution)
Every step in the proof can be justified except for Step 11. In Euclidean
geometry, there is always a unique circle passing through three noncollinear
points A, B, and C. That circle can be obtained by
finding the circumcenter of the triangle spanned by the three points. This
is the point equidistant from A, B, and C, obtained
by finding the common intersection of the perpendicular bisectors of the
sides of triangle ABC. Note that by construction, our given line
k
is the perpendicular bisector of AB, and line PR is the perpendicular
bisector of AC. By changing our choice of point
R, however,
it is possible to ensure that these two perpendicular bisectors do not
intersect, so that there is no circumcenter, and hence the circle does
not exist.
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