ISL 1979 Problem 79

Let $S$ be a unit circle and $K$ a subset of $S$ consisting of several closed arcs. Let $K$ satisfy the following properties: $(\text{i})$ $K$ contains three points $A,B,C$ , that are the vertices of an acute-angled triangle $(\text{ii})$ For every point $A$ that belongs to $K$ its diametrically opposite point $A'$ and all points $B$ on an arc of length $\frac{1}{9}$ with center $A'$ do not belong to $K$ . Prove that there are three points $E,F,G$ on $S$ that are vertices of an equilateral triangle and that do not belong to $K$ .