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Inscribed Angle Action! (2 Theorems)

Definition: An INSCRIBED ANGLE of a circle is an angle whose vertex lies on the circle and has each of its rays intersect the circle at one other point. Notice how both the inscribed angle and central angle both intercept the same arc. Before moving the slider on the right, feel free to move vertex of the inscribed angle (blue point) around anywhere you'd like. After dragging the slider (and future slider that will soon appear after that), be sure to answer any question(s) that may appear along the way. Have fun with this! Please thoroughly and specifically answer the questions that appear as best you can.