Secant Angles in a Circle
Secant is another name for a transversal. It's a straight line that crosses another shape of interest. Secants that cross a circle have many different properties. Below, our concern is with the angles formed by two secants that meet in a point, say, A. Such angles are appropriately called secant angles. With the exception of the case where A lies on the circle, the secant angle intercepts two arcs on the circle: BC and DE, in the applet. The secant angle equals half the sum of the angular measure of the two arcs (for A inside the circle) or half their difference (for A outside the circle). In the case where one of the secants is tangent to the circle, A still lies outside the circle and the angle intercepts two arcs. The angle is half their difference, as before.
The exceptional case, where A lies on the circle, happens to be the most fundamental. It is treated separately and underlies the proofs for the other cases.
What if applet does not run? 
Add one ancillary line to the diagram: DC. This creates a triangle ACD with an external angle BDC, for A outside the circle, and external angle BAC, for A inside the circle. In both cases, the inscribed angles BDC and DCE subtend the intercepted arcs. The result then follows from the Exterior Angle Theorem:
A is outside the circle  A is inside the circle  



With the notion of the directed angle, the two cases reduce to one. When A is outside the circle, the arcs BC and DE have different orientations, and the corresponding inscribed angles have different signs. For A inside the circle, the signs are the same.
A special case where, say, AB is tangent to the circle, so that B and D coalesce, while C lies opposite to B across the circle, deserves special attention. In this case, ΔABC, by construction, while ΔBEC is right, because angle BEC subtends a diameter of the circle. The angles ABE (formed by a tangent and a secant at B) and BCE are equal, which allows us to conclude that the former has the same measure as the arc BE enclosed by its sides: the tangent AB and the secant BE.
(A different approach appears elsewhere.)
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