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Let Sε denote the set of Euclidean triangles whose two small angles are within ε radians of 
and 
respectively. In this paper we prove two complementary theorems: (1) For any ε > 0 there exists a triangle in Sε that has no periodic billiard path of combinatorial length less than 1/ε. (2) Every triangle in S
1/400 has a periodic billiard path.
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