  We show that length minimizing curves in Carnot-Carath\'eodory spaces possess
at any point at least one tangent curve (i.e., a blow-up in the nilpotent
approximation) equal to a straight horizontal line. This is the first
regularity result for length minimizers that holds with no assumption on either
the space (e.g., its rank, step, or analyticity) or the curve, and it is novel
even in the setting of Carnot groups.
