What is a proof that a diameter bisects a circle?

Question

One of the major contributions Thales is said to have given is the proof that a diameter of a circle bisects the circle, yet Euclid doesn't even bat an eye. Then again, Euclid skipped over other things like needing to assume that the plane was complete.

First off, what do you think Thales meant by 'circle'? What about 'bisect'?

Second, how would you give a formal proof? The closest answer I've found by searching is at https://proofwiki.org/wiki/Circle_is_Bisected_by_Diameter

and the top is the most confusing proof I have ever seen.

Answer 1

See: Elements I.Def.18 :

A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.

We do not know that the semicircle is "half" of a circle.

See III.Prop.31 :

In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle.

See III.Def.11:

Similar segments of circles are those which admit equal angles, or in which the angles equal one another.

Therefore, the two semicircles of a circle are similar segments.

Then we need III.Prop.24 :

Similar segments of circles on equal straight lines equal one another.

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See :

• Robert Simson, (181) The Elements of Euclid, viz. the First Six Books, Together with the Eleventh and Twelfth. The Errors, by which Theon, Or Others, Have Long Ago Vitiated These Books are Corrected, and Some of Euclid's Demonstrations are Restored, page 297 : Note on Def.XVII, Bk.I.