TITLE: How to prove these two "obvious" facts about circles?
QUESTION [5 upvotes]: 1) It's said that Thales proved that "a diameter divides a circle in two congruent parts". I searched a lot but did not find his proof. How can it be proved?
2) How to prove that "a circle is a convex figure"?
These are things that can be easily noted by drawing a circle and looking at it... But is it possible to prove them? For example, I found in a book that one can use the triangle inequality to show that the diameter is the largest chord... Everyone knows that, but I found it very interesting that there was a proof for that! :) Is there any book that discusses "proving the obvious"?
Thanks!

REPLY [4 votes]: This is so basic that I’m not sure I can give a convincing answer. But the primitive notion of congruence between two Euclidean figures is that there should be a rigid motion of the plane (also allowing reflections) that transforms the first figure into the other. @Arash’s perfectly acceptable answer to your first question used a reflection, but let me give another answer.
Consider the $180^\circ$ rotation about the center of the circle. It transforms your specified diagonal into itself, since the diagonal is a line passing through the pivot-point center. And it transforms each point on the circumference of the one semicircle to a point on the circumference of the other semicircle. Similarly, points within the one semicircle get sent to interior points of the other. That should do it, and I think that Thales would recognize this as a proof.