Circle is Bisected by Diameter

Theorem

A circle is bisected by a diameter.


Proof 1

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

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

Aiming for a contradiction, suppose 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:

A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;
And the point is called the center 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.

$\blacksquare$


Proof 2

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

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

$\angle AOB\cong\angle BOA$ because they are both straight angles.

Thus, the arcs are congruent by Equal Angles in Equal Circles:

$\stackrel{\frown}{AB}\cong\stackrel{\frown}{BA}$

Hence, a circle is split into two equal arcs by a diameter.

$\blacksquare$


Historical Note

The result that a was supposedly attributed to Thales of Miletus by Proclus Lycaeus.

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

A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center.


According to Julian Lowell Coolidge in his A History of Geometrical Methods:

It seems strange that Thales, a disciple of the Egyptians, should bother to demonstrate something which Euclid takes as self-evident. Proclus suggests that he proved it by folding the circle over the diameter.


Sources

  • 1921: Sir Thomas Heath: A History of Greek Mathematics: Volume $\text { I }$ ... (previous) ... (next): $\text I$: Introductory: The Greeks and Mathematics
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