[color=#ff0000]In the hyperbolic plane[/color], given two circles with their center and one circumpoint, construct their common internal and external tangents

This exercise is well known in Euclidean geometry even as a classroom activity. However, the solution is not simple at all—it is based on the fact that for each two circles in the Euclidean plane there is a similarity transformation—namely, stretching w.r.t. a center—to map the first circle to the second one. If the circles are not concentric and not congruent, they have two similarity points. [br][br]In hyperbolic geometry, similarity (and stretching w.r.t. a center) is meaningless. Hence solving this problem in the P-model is a real challenge.

First we need to construct the centers of the pencils that contain the external and internal tangent lines.[br]Actually, this task [url=https://www.geogebra.org/m/ck6ecca5#material/upyvmnfq]has already been solved[/url]. There we learned that the center of the pencil could be constructed from two arbitrary elements of the pencil. One of them could be the line joining the centers of the circles as a free choice.[br][br]The point of concurrency for the internal tangents is a point, but in case one of the circles is big enough compared to the other one, then the center of the pencil of the external tangents can also be a point. In this case [url=https://www.geogebra.org/m/ck6ecca5#material/err565zq]Euclid's construction [/url] can be used to construct the tangents. In case the external tangents are a pair of ultraparallel lines, then the center of the pencil is a line, and the sought tangents are perpendicular to that line. We already found [url=https://www.geogebra.org/m/ck6ecca5#material/bmdyhhag]an example of such a construction[/url].[br][br]In the solution sketched up above we referred to three other problems which were previously discussed. What is more, one of them was solved by Euclid, and another one—indirectly—[url=https://www.geogebra.org/m/ck6ecca5#material/cgjmdhsj]by János Bolyai[/url]. Hopefully the reader agrees that this confirms the beauty of geometry.[br]