Here is an unusual proof - but perhaps easy to 'make sense of'.

Take the center of the circle.
Do a half-turn about this center.
- the points on the circle go to points on the circle.
- the diameter goes onto the diameter.
- the two tangents trade places.

IF we assume the two tangents originally meet at a point p,
then after the half-turn, they now meet at the the half-turn image of p - call it q.
We conclude that the two lines actually meet at two points p and q.

Two lines which meet at two points (or more) are the same line! Contradiction.

We conclude the lines did not meet, and are parallel.

Walter Whiteley