Reflecting a triangle in the plane

We prove that if the three angles of a triangle T in the plane are different from (60~176176 (30 ~ 30 ~ 120~ (45~176176 ~ 60~176 then the set of vertices of those triangles which are obtained from T by repeating 'edge-reflection' is everywhere dense in the plane. Introduction An edge-reflection of a triangle T 1 is a triangle T2 which is symmetric to T 1 with respect to the line determined by an edge of 7"1 (see Fig. 1). By a chain of triangles we mean a sequence of triangles T1, T2, T3 . . . . such that T~ (i > 2) is an edge-reflection of T~_~, and T~ ~ T~_2 for i > 3. Two triangles ABC and PQR are equivalent to each other if ABC = PQR or there is a finite chain of triangles T~ . . . . . T~ such that 7"1 = ABC and T~ = PQR. This is clearly an equivalence relation. Let us denote by f2as c (or simply by g2) the set of vertices of the triangles equivalent to a given triangle ABC. Figure 2 shows part of f2 for four types of triangles with angles (60 ~ 60 ~ 60~ (30 ~ 30 ~ 120~ (45 ~ 45 ~ 90~ (30 ~ 60 ~ 90~ We are going to prove that except for the above four types of triangles, ~ is everywhere dense in the plane (Theorems 2 and 3). The Angles of a Triangle In this paper all angles are measured by degree (o). A triangle ABC is called rational if its three angles are all rational angles, otherwise, ABC is called irrational. It is obvious that if ABC is irrational, then at least two angles are irrational. 98 I. B/irfiny et al.