Circle is Bisected by Diameter

Theorem
A circle is bisected by a diameter.

Proof

 * CircleBisectedByDiameter.png

Let $AB$ be a diameter of a circle $ADBE$ whose center is at $C$.

By definition of diameter, $AB$ passes through $C$.

that $AB$ does not bisect $ADBE$, but that $ADBC$ is larger than $AEBC$.

Thus it will be possible to find a diameter $DE$ passing through $C$ such that $DC \ne CE$.

Both $DC$ and $CE$ are radii of $ADBE$.

By Euclid's definition of the circle:

That is, all radii of $ADBE$ are equal.

But $DC \ne CE$.

From this contradiction it follows that $AB$ bisects the circle.

Historical Note
This was supposedly attributed to by.

defines the diameter as the line which passes through the center, but then assumes that it necessarily bisects it:



According to in his :
 * It seems strange that, a disciple of the Egyptians, should bother to demonstrate something which takes as self-evident.  suggests that he proved it by folding the circle over the diameter.

The proof given above is probably not the one given by, but is a product of the author of this page.