How I prove the underline text:

A line that touch only with one point a circle called **tangent**.__ By the symmetry property of the circle we can show that the tangent is normal (vertical) to radius in the point of tangent.__

What are the meaning of the property of symmetry and the __connection__ to the tangential line?

There are probably many ways you could use symmetry in such a proof; I'm not sure what the author specifically had in mind. Halls' idea, I think, is that if you reflect the entire figure (circle and tangent line) over the line through the center O and the point of tangency C, it must remain unchanged (that's what symmetry means); so the angles on each side at C must be equal. That implies they are both right angles.

You could also approach this as a proof by contradiction, by supposing that the line is

*not *perpendicular to the radius, doing the reflection, and showing that the line would not be tangent (because it would have to intersect the circle in two points).

If you want help figuring out what proof the author was thinking of, perhaps that would be clearer if we saw more of the context, including the explicit statement of the symmetry property of the circle, and any other examples of how the author uses symmetry. That's what I'd be looking at if I were you.

Incidentally, I would say "the tangent is perpendicular to the radius"; the word "normal" is valid, but I'd use it in different contexts, while "vertical" is not appropriate here.