Obtuse Triangular Billiards I: Near the (2,3,6) Triangle to appear

Let Sǫ denote the set of Euclidean triangles whose two small angles are within ǫ radians of π6 and π 3 respectively. In this paper we prove two complementary theorems: • For any ǫ> 0 there exists a triangle in Sǫ which has no periodic billiard path of combinatorial length less than 1/ǫ. • Every triangle in S1/400 has a periodic billiard path.