Third circle theorem proof
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'Angles in the same segment are equal'
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The 2 angles D and E are both sitting on the same piece of the circle, BC. ('Subtending the same minor arc BC' is a more mathsy way of putting it!).
In the second diagram, with the chord BC drawn, it is more obvious that angle D and E are in the same segment. It is also clear that A, the 'angle at the centre' BAC, is 'looking at' the same arc BC as the angles D and E.
So Theorem 1 applies: 'the angle at the centre is twice the angle at the circumference';
A=2D and A=2E, thus D=E.
ie, 'Angles in the same segment are equal'