Tangent to curves

 From:  bemfarmer
Is there a way to find the tangent Point, between a curve, and a given vector indicated by a line? (or a script.)

For example, the minimum of the curve, indicated by a horizontal vector (line), near the red arrow.

Or for example, the rightmost point of the curve, indicated by the vertical vector (line), near the other red arrow.

(Placing two tan-tan circles between a line and the curve, at the two outer points where the radius gets minimized, narrows down the tangent point.)