The hyperbolic plane

This applet represents the Poincaré model of the hyperbolic plane, which corresponds to the white interior of the pictured circle.

You can explore many aspects of hyperbolic geometry, e.g.:

examine the sum of the interior angles of triangles observing, in particular, what happens when the sides of the triangle become very small;
given a point exterior to a line , construct the perpendicular to passing through , and then the perpendicular to passing through (this is the Euclidean construction of a parallel);
given a point exterior to a line , construct as before the perpendiculars and , select any point on , draw the perpendicular to passing through , and the circle with center and radius , where is the intersection of and . Name the point of intersection of and which lies between and , and draw the line passing through and . What can you say about ?

How many lines exist that are parallel to a given line

and pass through an exterior point

? The latter is the János Bolyai construction of the asymptotic parallel line (there are two such lines for any

and

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