The chord angle theorem states that in an inscribed triangle (ABC) where A is the center of the circle and BC is a chord, and BDC is an inscribed triangle on the same chord, angle BDC must equal one half of angle BAC. Try changing the angle and moving point D and observe the theoremâ€™s truth.
Note: the measure of angle BDC is being constantly recalculated as point D is dragged, but it doesn't change because of this theorem.

Drag points A, B, and C to change the size and shape of the blue triangle, and its white counterpart that is similar (constrained by proportional SAS). Drag the Red point D to change the ratio in sizes.
Observe the multitude of calculated output lengths and angles, and how they match the proportion value, proving similarity, regardless of the triangles' shapes/sizes.

The Cyclic Quadrilateral Theorem states that for a quadrilateral inscribed in a circle, the measures of opposite angles must add to 180 degrees. Drag the points and observe the angle measures to see how this theorem holds true.