The inscribed angle theorem is a prerequisite needed to follow Ptolemy's proofs. It is a key for gaining an understanding of the heavens as he saw them. So it would not hurt to do a little review concerning inscribed angles. To prove the inscribed angle theorem we need the following figure.
In the figure the inscribed angle is angle ACB, its central angle is θ and the corresponding elements are indicated in green. Inside the circle are lines from its center to the ends, A and B, of the arc of the central angle and an additional line from the center to point C is needed to form two isosceles triangles inside the circle. In the following the inscribed angle is φ = γ + δ.
Since the inscribed angle, φ, is independent of α and β its value is the same for all inscribed angles on the same side of the of the chord, AB. The same half angle formula works for the remaining arc of the circle and its inscribed angle represented by the yellow lines. In this case the inscribed angle is again drawn on the side of the common chord opposite to that of the arc. The values of the two inscribed angles are not necessarily the same but since the sum of the two arcs of the circle is 360° the sum of their corresponding inscribed angles is 180°.