Circle theorems explorer – drag points, measure angles

What it shows

Eight standard circle theorems share one circle of radius 5 cm. Every angle and length is measured from the drawn points with vectors, not taken from the theorem, so each readout is an independent check. The angle at the centre is twice the angle at the circumference standing on the same arc, also when that central angle is reflex. Angles in the same segment are equal, while angles in opposite segments add up to 180°. A tangent is perpendicular to the radius, the two tangents from an external point are equal, and the perpendicular from the centre bisects a chord.

How to use

Choose a result in the Theorem list, then drag the white points on the circle, point M, or handle L, which turns the line about T. Untick Show the value to find so the class predicts the hidden angle or length before revealing it. Tick Proof hint to draw construction lines such as a diameter, radii or the foot of a perpendicular, together with the key step of the proof. Reset restores the starting figure.

Parameters you can change

• Theorem Angle at centre = 2 × angle at circumference, Angles in the same segment, Angle in a semicircle, Cyclic quadrilateral, Tangent ⟂ radius, Tangents from an external point, Alternate segment theorem, Perpendicular from centre to a chord
• Show the value to find
• Show construction lines and proof hint

Questions to explore

1. When P lies on the minor arc AB, which angle at the centre is twice angle APB?
2. How are angles APB and AQB related when P and Q lie in opposite segments?
3. Why does the line touch the circle at only one point when it is perpendicular to OT?