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.
(In reply to
Possible Solution by Harry)
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.

Posted by Bractals
on 20130512 19:55:54 