ILL 1986 Problem 20

For any angle α with $0 < \alpha < 180^{\circ}$ , we call a closed convex planar set an $\alpha$ -set if it is bounded by two circular arcs (or an arc and a line segment) whose angle of intersection is $\alpha$ . Given a (closed) triangle $T$ , find the greatest $\alpha$ such that any two points in $T$ are contained in an $\alpha$ -set $S \subset T .$