Inscribed Angle Theorem/Proof 1

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Theorem

An inscribed angle is equal to half the angle that is subtended by that arc.

InscribedAngleTheorem.png

Thus, in the figure above:

$\angle ABC = \frac 1 2 \angle ADC$

In the words of Euclid:

In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.

(The Elements: Book $\text{III}$: Proposition $20$)


Proof

Euclid-III-20.png

Let $ABC$ be a circle, let $\angle BEC$ be an angle at its center, and let $\angle BAC$ be an angle at the circumference.

Let these angles have the same arc $BC$ at their base.


Let $AE$ be joined and drawn through to $F$.

Since $EA = EB$, then from Isosceles Triangle has Two Equal Angles we have that $\angle EBA = \angle EAB$.

So $\angle EBA + \angle EAB = 2 \angle EAB$.

But from Sum of Angles of Triangle equals Two Right Angles we have that $\angle BEF = \angle EBA + \angle EAB$.

That is, $\angle BEF = 2 \angle EAB$.

For the same reason, $\angle FEC = 2 \angle EAC$.

So adding them together, we see that $\angle BEC = 2 \angle BAC$.


Now consider the point $D$, from which we have another angle $\angle BDC$.

Let $DE$ be joined and produced to $G$.

Similarly, we prove that $\angle GEC = 2 \angle EDC$.

Then $\angle GEB = 2 \angle EDB$.

Therefore $\angle BEC$ which remains is equal to $2 \angle BDC$.

Hence the result.

$\blacksquare$


Historical Note

This theorem is Proposition $20$ of Book $\text{III}$ of Euclid's The Elements.


Sources