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0401 Remarkable points and lines in a triangle

Autor:szilassilajos@t-online.hu
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.  
Let’s draw a triangle using its vertices! Let’s construct its sides, and then illustrate the following relationships in order:
  • If the perpendiculars drawn from the midpoints of any two sides of a triangle intersect, then the perpendicular drawn from the midpoint of the third side also passes through that point of intersection. This point is the center of the circle circumscribed around the triangle.
It is important to draw our readers’ attention to the condition of the above statement.
  • It can be proven using the Euclidean parallel postulate that the perpendiculars to the midlines of any two sides of a triangle intersect.
In hyperbolic geometry, there are triangles whose perpendicular bisectors of the sides do not intersect. We will discuss this in more detail later.
  • Let us show that we can formulate statements similar to the three above regarding the altitudes of a triangle!
Let’s construct the medians of the triangle! (To do this, we need to draw the perpendiculars to the midpoints of the sides. The points of intersection of these perpendiculars and the sides of the triangle will be the midpoints of the sides—just as in Euclidean geometry.)
  • Let’s show that: the medians of the triangle intersect at a single point!
If we think about it a little more carefully, it is surprising that this is an absolute geometric statement, since well-known relationship from Euclidean geometry was proven in school using similarity, and indirectly, the Euclidean parallel postulate. It would certainly be much more difficult to prove the statement using the tools of absolute geometry.

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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.

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