0401 Remarkable points and lines in a triangle
By using dynamic geometry the entire H-line can be displayed, including the border of the disk. This opens up new horizons in abstraction. As already mentioned, Bolyai proved that Euclidean geometry is a limiting case of the hyperbolic geometry system. If the disk of the P-model is much larger than the visible construction we work with, we can obtain drawings similar to the Euclidean ones, by using the P-model. On
a much-enlarged disk it is hard to notice that we work in the P-model, just as with standing on the floor it is hard to notice that we live on a spherical planet.
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040102
Are the three perpendicular bisectors of a triangle concurrent? The affirmative answer “Yes!” seems so obvious in the Euclidean plane that the question does not come up in most cases. However, the visualization in the P-model shows that the answer is very far from trivial in another system of geometry, hyperbolic geometry. The answer may be negative “No!” in sharp contrast with the Euclidean case.