Is the minimum of short run average cost equal to the minimum of long run average cost?

I understand that for perfect competition, the price is equal to minimum short and long run average cost in the long run as there cannot be any supernormal profits.

Does this mean that the short run and long run average cost at their minimum are equal for perfect competition in the long run? My textbook states that they are “tangential” to one another, but shouldn’t they be equal if there are no supernormal profits?

A wikiedpia article states one STC curve is tangent to LRTC at the long–run cost–minimizing level of production. At the point of tangency LRTC = STC. At all other levels of production STC will exceed LRTC. Why is STC equal to LRTC at this point and not at others? And does this also refer to average costs?