Hyperbolic perpendicular line

The construction of hyperbolic perpendicular lines. As usual this is done via the intersection of two circles.
Hyperbolic perpendicular
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Hyperbolic midpoint

Construction of a hyperbolic midpoint. This works the same as in the euclidean case.
Hyperbolic midpoint
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Angle bisection

Construction of a angle bisecting line. This works the same as in the euclidean case.
Angle bisection
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Center of circle

The classical construction for finding the center point of a circle.
Center of circle
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Theorem of Thales

The theorem of Thales does not hold in the hyperbolic plane.
Theorem of Thales
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Equilateral triangle

An equilateral triangle construction.
Hyperbolic equilateral triangle
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Circumcircle of a hyperbolic triangle

The constructoin of a circumcircle does in principle work, but sometimes the the lines do not intersect thus the midpoint can not be found.
Circumcircle of a hyperbolic triangle
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Circumcircle of a hyperbolic triangle
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Incircle of a hyperbolic triangle

Construction of an incircle of an hyperbolic triangle works like in the euclidean case.
Incircle of a hyperbolic triangle
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Golden ratio

The usual construction of the golden ratio fails. Mathematically, there exists a golden ratio, but it can not be costructed this way.
Golden ratio
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Mirror circle

Mirroring a circle again and again along a line ...
Mirror circle
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Hyperbolic regular pentagon

Two similar constructions of hyperbolic regular pentagons. Both constructions fail, as the diagonals can not be constructed by means of the golden ratio as it is done in the euclidean case.
Hyperbolic regular pentagon
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Hyperbolic regular pentagon
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Hyperbolic regular hexagon

The classical construction of an hyperbolic hexagon fails too. The well known formula U = 2*pi*r for circles has to be modified for hyperbolic circles to U = 2*pi*sinh(r).
Hyperbolic regular hexagon
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$Update: Sun, 26 May 2013 20:16:05 +0200 $, $Revision: 9c1955f3e6d5 $