1.4142 = square root of 2

Walk 1.4142km from the tree. Turn 45 degrees left, then walk 1km.
Follow an arc with the radius of 1km from the tree for 3.1416km
Walk 1km straight from the point you stopped.

This is a horseshoe shaped path.

Distance = 6.556km

This path will intersect all tangents of the circle (worst case what we are looking for, I'm guessing). That's as good as I can do! Please post the real solution when you find it, Mark!