10

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.

MJD
  • 67,568
  • 43
  • 308
  • 617
Aaron Goldsmith
  • 660
  • Is it not a definition of a property of a line through a circle? Rather than saying this line through this circle is a diameter... let's check if it passes through the centre. It's a case of ... this line passes through the centre of this circle... therefore it's a diameter. ? Does it require proof? – Fogmeister Jan 09 '17 at 17:59

1 Answers1

5

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.

See :

Mauro ALLEGRANZA
  • 99,247