Right Angle to Tangent of Circle goes through Center

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Theorem

In the words of Euclid:

If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the tangent, the centre will be on the straight line so drawn.

(The Elements: Book $\text{III}$: Proposition $19$)


Proof

Euclid-III-19.png

Let $DE$ touch the circle $ABC$, and let $AC$ be drawn at right angles to $DE$.


Suppose the center is not on $AC$.

Let $F$ be the center of $ABC$, and let $FC$ be joined from there to the point of contact.

By Radius at Right Angle to Tangent, $FC$ is perpendicular to $DE$.

Therefore $\angle FCE$ is a right angle.

But by hypothesis $\angle ACE$ is also a right angle.

Thus $\angle FCE = \angle ACE$, which is impossible unless $F$ lies on $AC$.


Therefore the center of $ABC$ lies on $AC$.

$\blacksquare$


Historical Note

This theorem is Proposition $19$ of Book $\text{III}$ of Euclid's The Elements.


Sources