Perpendicular Bisector of Chord Passes Through Center

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Theorem

The perpendicular bisector of any chord of any given circle must pass through the center of that circle.


In the words of Euclid:

From this it is manifest that, if in a circle a straight line cut a straight line into two equal parts and at right angles, the centre of the circle is on the cutting straight line.

(The Elements: Book $\text{III}$: Proposition $1$ : Porism)


Proof

BisectorOfChord.png

Let $F$ be the center of the circle in question.

Draw any chord $AB$ on the circle.

Bisect $AB$ at $D$.

Construct $CE$ perpendicular to $AB$ at $D$, where $D$ and $E$ are where this perpendicular meets the circle.

Then the center $F$ lies on $CE$.


The proof is as follows.


Join $FA, FD, FB$.

As $F$ is the center, $FA = FB$.

Also, as $D$ bisects $AB$, we have $DA = DB$.

As $FD$ is common, then from Triangle Side-Side-Side Equality, $\triangle ADF = \triangle BDF$.


In particular, $\angle ADF = \angle BDF$; both are right angles.

From Book $\text{I}$ Definition $10$: Right Angle:

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

So $\angle ADF$ and $\angle BDF$ are both right angles.

Thus, by definition, $F$ lies on the perpendicular bisector of $AB$.

Hence the result.

$\blacksquare$


Historical Note

The argument for this particular result originates from Proposition $1$ of Book $\text{III}$ of Euclid's The Elements.

However, the result itself is due to Augustus De Morgan, who reasoned that this result was more fundamental.


This theorem is the converse of Proposition $3$ of Book $\text{III} $: Conditions for Diameter to be Perpendicular Bisector.


Sources