How is a circle convex?

Answer 1

I'm not sure whether you are asking what convex means, or to prove that all circles are convex. A shape is convex if all pairs of points within it (i.e., the interior, not just the boundary) have all points between them within the shape. That is, if you draw the straight line segment between 2 points on or within a circle, all points of that line segment are within the circle.

To prove that, you could use Euclid's axioms of geometry, or analytic geometry, which deals with the coordinates of points and the equations they satisfy. I will leave the proof as an exercise, since it is rather involved (but not necessarily advanced) and I don't know if that is what you are asking for anyway!