Let D be a point in the plane of ΔABC such that
ray AD intersects the interior of side BC.

Construct a line through D intersecting lines AB and
AC at points B' and C' respectively such that the
perimeter of ΔAB'C' equals |BC|.

There are two to four solutions to the problem depending on the location of point D. When you say a line exists only if D is outside the circle - outside towards A or away from A? In your case, outside away from A has no solution and on the smaller arc RS has one solution. Check out my posted solution.