This Demonstration shows the surprising existence of a smooth strictly convex closed curve (an oval) for which it is possible to walk around the exterior so that the two tangent segments drawn from your vantage point are never the same length.
Viewing the oval from the outside, the left (red) tangent is always longer than the right (blue) tangent. The ratio of the length of the right tangent to the length of the left tangent is shown; it is at most 0.90. Snapshot 1: To better see that the red left tangent is always longer, click the "show circles" checkbox to see that the red circle with the left tangent radius always strictly contains the blue circle with the right tangent radius.
The star-shaped outer curve is the walker's path to give this result. It can be hidden by unchecking its box (shown in snapshot 1 and hidden in snapshot 3).
To see the smoothness of the oval more clearly, click the button for either the left edge or right edge showing a magnified view of the point of tangency of that segment. The curve is made up of circular arcs with matching tangency where connected, so the smoothness is of type C1 (continuously turning tangents); see snapshot 2.
The chord that connects the points of tangency is just to help follow the action; in a nonstrictly convex oval it would be necessary to make a choice of tangent points (shown in snapshot 3, hidden in snapshot 1).
 S. Tabachnikov, "The (Un)equal Tangents Problem," The American Mathematical Monthly, 2012 pp. 398–405.