Angles on Equal Arcs are Equal

Theorem
In equal circles, angles standing on equal circumferences are equal to one another, whether they stand at the centeror at the circumferences.

Proof
Let $ABC$ and $DEF$ be equal circles.

Let $\angle BGC$ and $\angle EHF$ stand on the equal circumferences $BC$ and $EF$.


 * Euclid-III-27.png

Suppose $\angle BGC \ne \angle EHF$.

Then one of them is bigger. Suppose $\angle BGC > \angle EHF$.

On the straight line $BG$, construct $\angle BGK$ equal to $\angle EHF$.

From Equal Angles in Equal Circles, these angles stand on equal circumferences.

So circumferences $BK = EF$.

But $EF = BC$ and so $BK = BC$, which is impossible.

So $\angle BGC$ and $\angle EHF$ are not unequal, therefore $\angle BGC = \angle EHF$.

From the Inscribed Angle Theorem, $2 \angle BAC = \angle BGC$ and $2 \angle EDF = \angle EHF$.

Therefore $\angle BAC = \angle EHF$.