Welcome to the Interactive Circle Theorems Tutorial! This page is designed to help you understand and visualize some of the most common and important theorems related to circles in geometry. Each section provides a clear explanation and an interactive simulation, allowing you to explore the theorem in action by dragging points (with mouse or touch) and observing the dynamic changes in angles and relationships.
Theorem 1: Angle at the Center
The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference.
Explanation: Imagine an arc (part of the circle's edge). The angle formed by joining the ends of this arc to the circle's center is the 'angle at the center'. If you pick another point on the circumference (not on the arc itself) and join it to the ends of the same arc, you form the 'angle at the circumference'. This theorem states the center angle is twice the circumference angle. This holds true whether you consider the minor arc or the major arc (where the angle at the center would be a reflex angle).
Drag points P1, P2 (to define the arc) and P_circ (the point on the circumference).
Angle at Center (P1-O-P2): 0°
Angle at Circumference (P1-P_circ-P2): 0°
Theorem 2: Angle in a Semicircle
The angle in a semicircle is always a right angle (90 degrees).
Explanation: A semicircle is half of a circle, formed by a diameter and the arc it cuts off. If you pick any point on the circumference of this semicircle and connect it to the two ends of the diameter, the angle formed at that point will always be 90 degrees. This is a special case of the 'Angle at the Center' theorem, where the angle at the center is a straight line (180°).
Drag point A to move the diameter, and point P along the circumference.
Angle APB: 0°
Theorem 3: Angles in the Same Segment
Angles subtended by the same arc at the circumference are equal, provided they are in the same segment.
Explanation: A chord (a line connecting two points on the circle) divides the circle into two segments (a major and a minor one). If you take an arc defined by this chord, any angle you form by picking a point on the circumference within one segment and connecting it to the ends of the arc will be the same value.
Drag points P1, P2 to define the arc/chord. Drag C1 and C2 (points on the circumference).
Angle at C1 (P1-C1-P2): 0°
Angle at C2 (P1-C2-P2): 0°
Theorem 4: Opposite Angles of a Cyclic Quadrilateral
The opposite angles of a cyclic quadrilateral (a quadrilateral whose vertices all lie on the circumference of a circle) are supplementary (they add up to 180 degrees).
Explanation: If you draw a four-sided shape (quadrilateral) such that all four of its corners (vertices) touch the circle's edge, then the sum of any pair of opposite angles will always be 180°. For example, if the angles are A, B, C, and D in order around the quadrilateral, then A + C = 180° and B + D = 180°.
Drag points A, B, C, and D on the circumference.
Angle at 1st vertex: 0° | Angle at 3rd vertex: 0° | Sum: 0°
Angle at 2nd vertex: 0° | Angle at 4th vertex: 0° | Sum: 0°
Theorem 5: Alternate Segment Theorem
The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
Explanation: Consider a circle, a tangent line touching the circle at a point (let's call this P2), and a chord drawn from P2 to another point on the circle (say P1). The angle formed between the tangent (at P2) and the chord P1P2 is equal to any angle subtended by the chord P1P2 at a point C on the circumference in the alternate segment. The "alternate segment" is the one that doesn't contain the angle formed by the tangent and the chord.
Drag points P1 (chord end), P2 (point of tangency & chord end), and C (on alternate segment).
Angle (Tangent at P2 - Chord P1P2): 0°
Angle in Alternate Segment (P1CP2): 0°
Theorem 6: Tangents from an External Point
Tangents drawn from an external point to a circle are equal in length. Also, the line joining the external point to the center of the circle bisects the angle between the tangents.
Explanation: If you pick a point outside the circle and draw two lines that just touch the circle (tangents), the lengths of these tangent segments (from the external point to the points of tangency) will be identical. Furthermore, the line connecting the external point to the circle's center will divide the angle formed by the two tangents into two equal angles. The radii to the points of tangency will also form 90° angles with the tangents.
Drag point P_ext (the external point).
Length of Tangent 1 (P_ext-T1): 0
Length of Tangent 2 (P_ext-T2): 0
Angle OP_extT1: 0°
Angle OP_extT2: 0°
Theorem 7: Perpendicular from Center to Chord
A line drawn from the center of a circle perpendicular to a chord bisects the chord. Conversely, the line joining the center of a circle to the midpoint of a chord is perpendicular to the chord.
Explanation: If you draw a line segment from the center of the circle (O) to meet a chord (AB) at a point (M) such that OM is perpendicular to AB, then M will be the midpoint of AB (meaning AM = MB). Conversely, if M is the midpoint of the chord AB, then the line segment OM will be perpendicular to the chord AB (forming a 90° angle). Our simulation defines M as the midpoint of chord AB. Observe that the line OM is then perpendicular to the chord AB, and the segments AM and MB are equal.
Drag points A and B on the circumference to change the chord.
Length AM: 0
Length MB: 0
Angle OMA: 0°
We hope these interactive simulations have helped you grasp the concepts behind these circle theorems. The best way to learn is by doing! We encourage you to experiment with all the draggable points in each simulation. Try different configurations, move points to extreme positions, and observe how the angles and relationships change—or stay constant—according to the theorem. Happy learning!