By F. Oort

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Sinhp2. 2 Proof. We give a proof which is based on a decomposition into trirectangles and right-angled triangles. 7. 2, §3] Trirectangles and Pentagons 39 We may assume that pl > 0. Dropping the perpendicular r from/?! to that geodesic through p2 which is orthogonal to 77, we obtain a trirectangle with sides ply\t2-t1\tsi say, and r and also a right-angled triangle with sides r, \p2- s\ and c, whose hypothenuse c has length dist(pl,p2). The trirectangle is again decomposed into two right-angled triangles with common hypothe nuse.

7 Theorem. Let n > 0 be an integer and consider two non-trivial closed curves c and c' ona compact hyperbolic surface S which intersect each other in n points. Ify and y' are the closed geodesies in their free homotopy classes, then either y and y' coincide as point sets or they intersect each other in at most n points. Proof. We prove the first theorem and leave the proof of the second theorem as an exercise. 2 with the Arzela-Ascoli argument. The compactness of S is needed to assure that the sequence { yn }™=l stays within bounded distance.

3 Definition. l, §6 Sl of the form h(t) = t + const such that y' = y ° h. A closed geodesic is an equivalence class of closed parametrized geodesies. Note that equivalent closed geodesies are always freely homotopic. Length, orientation and the property of being homotopic to other closed curves carry over from the parametrized closed geodesies in a natural way since these quantities are invariant under the above homeomorphisms. If no confusion arises, we identify closed geodesies with their parametrized representatives.