Circle Theorems Calculator

Interactive circle theorems calculator. Explore inscribed angles, tangent-chord angles, intersecting chords, secant-tangent relationships, and more.

Select Theorem

The inscribed angle is half the central angle that subtends the same arc.

Result

Inscribed Angle
60
degrees

Step-by-Step Solution

Inscribed Angle = Central Angle / 2

Understanding Circle Theorems

Circle theorems are a set of fundamental rules governing the relationships between angles, arcs, chords, tangents, and secants of a circle. They are essential in geometry and are widely used in proofs, constructions, and real-world applications such as engineering, optics, and computer graphics.

Key Circle Theorems

Inscribed Angle Theorem

An inscribed angle is half the central angle that subtends the same arc.

Inscribed = Central / 2

Tangent-Chord Angle

The angle between a tangent and chord equals half the intercepted arc.

Angle = Arc / 2

Intersecting Chords Angle

The angle formed equals half the sum of the two intercepted arcs.

Angle = (Arc1 + Arc2) / 2

Intersecting Chords Segments

Products of the segments of each chord are equal.

a x b = c x d

Secant-Tangent Power

Tangent squared equals the product of secant external and whole length.

tan² = ext x whole

Two Secants Angle

Angle between two secants from an external point equals half the difference of intercepted arcs.

Angle = |Arc1 - Arc2| / 2

Detailed Theorem Explanations

Inscribed Angle Theorem

The Inscribed Angle Theorem states that an angle inscribed in a circle is half the central angle that intercepts the same arc. An inscribed angle is formed by two chords that share an endpoint on the circle. The vertex of an inscribed angle lies on the circle, while the vertex of a central angle is at the center. A key corollary is that all inscribed angles subtending the same arc are equal, regardless of where the vertex is placed on the major arc.

Tangent-Chord Angle Theorem

When a tangent line and a chord meet at a point on the circle, the angle between them equals half the intercepted arc. This theorem bridges the relationship between tangent lines (which touch the circle at exactly one point) and chords (which connect two points on the circle). The tangent-chord angle can be thought of as a limiting case of the inscribed angle where one side becomes tangent to the circle.

Intersecting Chords Theorem

When two chords intersect inside a circle, two important relationships hold. First, the angle formed equals half the sum of the two intercepted arcs. Second, the products of their segments are equal: if chord AB intersects chord CD at point P, then AP x PB = CP x PD. This segment relationship is a consequence of similar triangles formed by the intersecting chords.

Secant-Tangent Theorem (Power of a Point)

When a tangent and a secant are drawn from the same external point, the tangent length squared equals the product of the secant's external segment and its total length. This is a special case of the Power of a Point theorem. If the tangent has length t, the secant has external segment e and total length s, then t² = e x s.

Two Secants from an External Point

When two secants are drawn from the same external point, two relationships apply. The products of the external segment and whole length are equal for both secants. Also, the angle between the two secants equals half the absolute difference of the intercepted arcs. These relationships are fundamental in solving complex circle geometry problems.

Applications of Circle Theorems

  • Architecture: Designing arches, domes, and circular structures using angle and arc relationships.
  • Optics: Understanding how light reflects and refracts in circular lenses and mirrors.
  • Computer Graphics: Computing intersections, reflections, and tangent points for rendering circular objects.
  • Navigation: GPS triangulation and radar systems rely on circle intersection theorems.
  • Astronomy: Calculating orbital positions and eclipse geometry involves circle theorems.