About

Misapplying a circle theorem collapses an entire geometric proof. The inscribed angle theorem states that an inscribed angle equals exactly half the central angle subtending the same arc, i.e. θ_ins = 12 θ_cen. Confusing this with the tangent-chord angle or forgetting the cyclic quadrilateral constraint (α + γ = 180°) leads to wrong segment lengths and failed constructions. This calculator implements 12 standard circle theorems covering angle relationships, chord powers, and tangent-secant identities. It assumes Euclidean plane geometry with a perfect circle of radius r > 0.

Each theorem produces an interactive Canvas diagram so you can visually verify the geometric configuration before relying on the numerical output. Chord length uses c = 2r sin(θ⁄2) and the perpendicular bisector distance is derived via the Pythagorean theorem. All angle inputs are in degrees. Note: the tool does not handle degenerate cases where a chord length exceeds the diameter or where points coincide.

Formulas

The inscribed angle theorem is the foundation of most circle angle relationships:

θ_inscribed = 12 × arc_intercepted

For two chords intersecting inside the circle at point P, the angle and power of a point are:

θ = 12(arc₁ + arc₂)

a ⋅ b = c ⋅ d

For two secants from an external point, the angle uses the arc difference:

θ = 12 |arc_far − arc_near|

Chord length from central angle θ and radius r:

c = 2r sin(θ2)

Perpendicular distance from center to a chord of length c:

d = √r² − (c2)²

Where: r = radius, θ = central angle in degrees (converted to radians internally via θ_rad = θ_deg × π ⁄ 180), c = chord length, d = perpendicular distance, a, b, c, d = chord segment lengths, e = external segment, w = whole secant length, t = tangent length.

Reference Data

Theorem	Configuration	Key Relationship	Conditions
Inscribed Angle	Vertex on circle	θ_ins = 12 ⋅ arc	Vertex on circumference
Central Angle	Vertex at center	θ_cen = arc	Both radii to arc endpoints
Thales’ Theorem	Diameter as chord	θ = 90°	Subtended by diameter
Cyclic Quadrilateral	4 vertices on circle	α + γ = 180°	All vertices concyclic
Tangent-Radius	Tangent at point P	Angle = 90°	Tangent meets radius at P
Alternate Segment	Tangent + chord at P	Tangent-chord angle = inscribed angle in alt. segment	Chord from tangent point
Intersecting Chords	Two chords cross inside	a⋅b = c⋅d	Intersection inside circle
Intersecting Chords (Angle)	Two chords cross inside	θ = 12(arc₁ + arc₂)	Vertically opposite arcs
Secant-Secant (Angle)	Two secants from ext. point	θ = 12\|arc₁ − arc₂\|	External vertex
Secant-Secant (Power)	Two secants from ext. point	e₁⋅w₁ = e₂⋅w₂	ext × whole segments
Tangent-Secant	Tangent + secant from ext.	t² = e⋅w	Tangent length squared
Two Tangents	Two tangents from ext. point	Lengths equal; θ = 12\|major − minor\|	External vertex
Chord Length	Chord with central angle	c = 2r sin(θ⁄2)	θ in radians internally
Arc Length	Arc with central angle	s = rθ	θ in radians internally
Perpendicular to Chord	Center to chord distance	d = √r² − (c⁄2)²	c ≤ 2r

Frequently Asked Questions

They are equal regardless of where their vertices lie on the major arc. This is a direct corollary of the inscribed angle theorem: since both equal half the intercepted arc, they must be congruent. The calculator uses this principle - if you enter the same intercepted arc for two inscribed angles, both outputs will match.

As the intersection point approaches the circumference, one segment on each chord approaches zero length. The product remains constant and equal to the power of the point with respect to the circle. At the boundary the power equals zero. The formula a⋅b = c⋅d still holds, but with one factor being zero on each side.

Vertex position determines the formula. When the vertex is inside the circle, the angle opens between two arcs that together span the full intercepted region, so the angle averages the sum: θ = ½(arc₁ + arc₂). When the vertex is outside, the angle is formed by the gap between the far arc and the near arc: θ = ½|arc_far − arc_near|. This distinction is purely geometric - the sign flip comes from whether the vertex lies inside or outside the circle.

No. Since c = 2r sin(θ⁄2) and sin is bounded by 1, the maximum chord length is 2r, which occurs when θ = 180° (the chord is a diameter). The calculator validates this and will flag an error if you somehow input parameters that imply a chord exceeding the diameter.

A cyclic quadrilateral inscribed in a circle is necessarily convex. If the vertices are concyclic, no vertex can be inside the triangle formed by the other three. The supplementary opposite angles condition (α + γ = 180°) is both necessary and sufficient for a quadrilateral to be cyclic (Ptolemy's converse). If your measured angles do not sum to 180°, the quadrilateral is not inscribable in a circle.

The perpendicular distance d = √(r² − (c⁄2)²) decreases as the chord length increases. A longer chord sits closer to the center. When c = 2r (diameter), d = 0. When c approaches 0, d approaches r. This is a direct application of the Pythagorean theorem to the right triangle formed by the radius, half-chord, and perpendicular distance.