What are the 8 Circle Theorems in Euclidean Geometry?

The 8 primary circle theorems are: (1) Angle at the center is twice the angle at the circumference; (2) Angle in a semicircle is a right angle (90°, Thales's theorem); (3) Angles in the same segment subtended by the same arc are equal; (4) Opposite angles in a cyclic quadrilateral sum to 180°; (5) The angle between a tangent and radius is 90°; (6) Tangents from an external point are equal in length; (7) Alternate Segment Theorem (tangent-chord angle equals opposite inscribed angle); and (8) Intersecting Chords / Secant Theorem (AP * PB = CP * PD).

What is the Inscribed Angle Theorem formula?

The Inscribed Angle Theorem states that the central angle (theta_center) subtended by an arc is exactly twice the inscribed angle (theta_inscribed) subtending the same arc: theta_center = 2 * theta_inscribed, or conversely, theta_inscribed = theta_center / 2.

What is the Cyclic Quadrilateral Theorem?

In any four-sided polygon whose vertices all lie on a single circle (a cyclic quadrilateral), the opposite interior angles are supplementary (sum to 180°): angle A + angle C = 180° and angle B + angle D = 180°. Additionally, any exterior angle equals the opposite interior angle.

What is the Intersecting Chord Theorem formula?

When two chords AB and CD intersect inside a circle at point P, the products of their segments are equal: AP * PB = CP * PD. If you know three segment lengths, the fourth is calculated as PD = (AP * PB) / CP.

What is the Tangent-Secant Theorem formula?

If a tangent segment PT and a secant line passing through points A and B are drawn from an external point P, the square of the tangent length equals the product of the external secant segment and the entire secant segment: PT^2 = PA * PB PT = sqrt(PA * PB).

What is the Alternate Segment Theorem?

The Alternate Segment Theorem states that the angle between a tangent line and a chord passing through the point of contact is equal to the inscribed angle in the alternate (opposite) triangular segment: theta_tangent-chord = theta_alternate.

How do you calculate the angle formed by two intersecting chords inside a circle?

When two chords intersect inside a circle, the interior angle formed (theta_inside) is equal to half the sum of the two intercepted arcs: theta_inside = (Arc_1 + Arc_2) / 2.

How do you calculate the outside angle formed by two secants or tangents outside a circle?

When two secants, tangents, or a secant and tangent intersect outside a circle at point P, the exterior angle (theta_outside) is equal to half the difference of the intercepted arcs: theta_outside = (Arc_far - Arc_near) / 2.

What is the Gershgorin Circle Theorem in linear algebra?

The Gershgorin Circle Theorem states that for any square n x n complex matrix A, every eigenvalue of A lies within at least one closed circular disc in the complex plane centered at a_ii with radius R_i = sum_{j != i} |a_ij| (the sum of the absolute values of the off-diagonal entries in row i).

How does this Circle Theorems Calculator work and what problem types does it solve?

The calculator features interactive solvers for all 8 Euclidean circle theorems, intersecting chord and secant power-of-a-point equations, interior/exterior intercepted arc angles, cyclic quadrilateral area (Brahmagupta's formula), and Gershgorin eigenvalue disc bounds with instant step-by-step solutions.

100% Free • Euclidean Circle Theorems & Angle Solver

Circle Theorems Calculator

Solve inscribed angles ($\theta_{\text{center}} = 2\theta_{\text{inscribed}}$), cyclic quadrilaterals (opposite angles sum to $180^\circ$), intersecting chord segments ($AP \cdot PB = CP \cdot PD$), tangent-secant lengths, alternate segment angles, inside/outside intersection angles, and Gershgorin circle eigenvalue bounds.

Theorem Presets: Tap to load

Circle Theorem Solver

Select theorem rule and enter geometric parameters.

θ_center = 2θ_inscribed

Select Circle Theorem

Known Angle θ_c = 2θ_i

degrees (°)

Key Theorem Breakdown

Primary Output 60.00°

Theorem Law θ_i = θ_c / 2

Status Verified Exact

Inscribed Angle (θ_inscribed)

60.00°

Central Angle: 120.00° • θ_inscribed = 120.00° / 2 = 60.00°

Theorem Telemetry Matrix

📐 Solved Parameter	60.00°
⭕ Associated Angle / Value	Central Angle = 120.00°
📜 Applied Theorem	Inscribed Angle Theorem
🔢 Step-by-Step Proof	θ = 120.00° / 2 = 60.00°
💡 Supplementary / Extra	Major Arc = 240.00°

Why Use Our Circle Theorems Calculator? Real-World Problems It Solves

Euclidean circle theorems form the backbone of high school geometry (GCSE, IGCSE, SAT, AMC), surveying, celestial navigation, and structural engineering. However, identifying which theorem applies to intersecting lines, chords, or tangents can be challenging. Our Circle Theorems Calculator & Solver provides instant mathematical proof breakdowns:

1. Inscribed vs. Central Angle Verification:

Instantly verifies why the angle subtended at the center is always exactly twice the angle at the circumference ($\theta_{\text{center}} = 2\theta_{\text{inscribed}}$), and why Thales's semicircle angle is always an exact right angle ($90^\circ$).

2. Intersecting Chords & Power of a Point:

Solves unknown chord lengths inside circles ($AP \cdot PB = CP \cdot PD$) and external tangent-secant lengths ($PT^2 = PA \cdot PB$) without drawing complex similar triangle proofs.

3. Cyclic Quadrilateral Supplementary Angles:

Computes opposite interior angles ($\angle A + \angle C = 180^\circ$) and exterior angles for 4-sided polygons inscribed in circles.

4. Gershgorin Disc Bounds in Linear Algebra:

For advanced university mathematics and matrix eigenvalue bounds, calculates circular discs $D(a_{ii}, R_i)$ in the complex plane where all matrix eigenvalues are guaranteed to reside.

How to Solve Circle Theorems (Step-by-Step Guide)

Step 1: Choose Theorem

Select from Inscribed Angles, Cyclic Quad, Chords, Tangent-Secant, Arc Angles, or Gershgorin.

Step 2: Enter Known Values

Input given angle degrees (e.g. $120^\circ$) or line segment lengths (e.g. $AP=6, PB=4, CP=8$).

Step 3: Review Full Proof

Review the exact mathematical derivation, applied geometric rule, and supplementary arc measurements.

Step 4: Copy Proof Card

Click Copy Circle Theorem Proof Card to copy a formatted proof report to your clipboard.

The 8 Classical Euclidean Circle Theorems with Formulas

1. Angle at Center Theorem (Inscribed Angle)

The angle subtended by an arc at the center is twice the angle subtended at the circumference: $$\theta_{\text{center}} = 2 \times \theta_{\text{inscribed}} \iff \theta_{\text{inscribed}} = \frac{\theta_{\text{center}}}{2}$$

2. Thales's Theorem (Angle in a Semicircle)

An inscribed triangle whose base is the diameter always forms a right angle at the circumference: $$\theta_{\text{semicircle}} = 90^\circ$$

3. Angles in the Same Segment

Inscribed angles subtended by the same arc (or chord) are always equal: $$\alpha = \beta = \gamma$$

4. Cyclic Quadrilateral Theorem

Opposite interior angles of a cyclic quadrilateral always sum to $180^\circ$ (supplementary): $$\angle A + \angle C = 180^\circ, \quad \angle B + \angle D = 180^\circ$$

5. Radius-Tangent Perpendicularity

A tangent line to a circle is perpendicular to the radius at the point of contact: $$\angle(\text{Radius}, \text{Tangent}) = 90^\circ$$

6. Alternate Segment Theorem

The angle between a tangent and a chord equals the inscribed angle in the alternate segment: $$\theta_{\text{tangent-chord}} = \theta_{\text{alternate}}$$

7. Intersecting Chords Theorem

When two chords intersect inside a circle at point $P$: $$AP \cdot PB = CP \cdot PD \implies PD = \frac{AP \cdot PB}{CP}$$

8. Tangent-Secant Theorem (Power of a Point)

From external point $P$, tangent $PT$ and secant segments $PA, PB$ satisfy: $$PT^2 = PA \cdot PB \implies PT = \sqrt{PA \cdot PB}$$

Rigorous Mathematical Proofs of Core Circle Theorems

1. Proof of the Inscribed Angle Theorem ($\theta_c = 2\theta_i$):

Consider circle with center $O$ and chord $AB$ subtending central angle $\angle AOB$ and inscribed angle $\angle APB$. Draw line segment from $P$ through center $O$. Triangles $\triangle APO$ and $\triangle BPO$ are isosceles because $OA = OP = OB = r$ (circle radii).

• In $\triangle APO$, base angles are equal: $\angle OAP = \angle OPA = x$. By the exterior angle theorem, $\angle AOX = 2x$.

• In $\triangle BPO$, base angles are equal: $\angle OBP = \angle OPB = y$. By the exterior angle theorem, $\angle BOX = 2y$.

• Total inscribed angle is $\angle APB = x + y$.

• Total central angle is $\angle AOB = 2x + 2y = \mathbf{2(x + y)} = \mathbf{2 \times \angle APB}$. $\quad \blacksquare$

2. Proof of the Intersecting Chords Theorem ($AP \cdot PB = CP \cdot PD$):

Draw line segments $AC$ and $DB$. In triangles $\triangle APC$ and $\triangle DPB$:

• $\angle APC = \angle DPB$ (vertically opposite angles).

• $\angle PAC = \angle PDB$ (angles subtended by the same arc $CB$ in the same segment).

• By AA Similarity, $\triangle APC \sim \triangle DPB$.

• Therefore, corresponding sides are in proportion: $\frac{AP}{PD} = \frac{CP}{PB} \implies \mathbf{AP \cdot PB = CP \cdot PD}$. $\quad \blacksquare$

Classroom Practice & Study Guide: Circle Geometry Proofs

Worked Geometry Exam Problems with Solutions:

Problem 1: Inscribed Angle & Central Angle

An arc subtends a central angle of $110^\circ$. Find the angle subtended by this same arc at any point on the circumference.

Solution: $\theta_{\text{inscribed}} = \frac{110^\circ}{2} = \mathbf{55.00^\circ}$ (Inscribed Angle Theorem).

Problem 2: Intersecting Chords Segment Unknown

Two chords intersect inside a circle at $P$. If $AP = 6\,\text{cm}$, $PB = 4\,\text{cm}$, and $CP = 8\,\text{cm}$, calculate the length of $PD$.

Solution: $AP \cdot PB = CP \cdot PD \implies 6 \times 4 = 8 \times PD \implies 24 = 8 \times PD \implies PD = \mathbf{3.00\,\text{cm}}$.

Problem 3: Outside Angle from Intercepted Arcs

Two secants meet outside a circle. The far intercepted arc is $110^\circ$ and the near arc is $40^\circ$. Find the intersection angle.

Solution: $\theta_{\text{outside}} = \frac{110^\circ - 40^\circ}{2} = \frac{70^\circ}{2} = \mathbf{35.00^\circ}$.

Master Benchmark Circle Theorems Matrix Table

Comprehensive summary of Euclidean circle theorem formulas, inputs, and outputs.

Theorem Benchmarks

Theorem Name	Primary Formula	Sample Inputs	Calculated Result	Geometric Meaning
Inscribed Angle	$\theta_i = \theta_c / 2$	$\theta_c = 120^\circ$	$\theta_i = 60^\circ$	Center angle is 2x circumference
Thales Semicircle	$\theta = 90^\circ$	$\theta_c = 180^\circ$	$\theta = 90^\circ$	Right angle in semicircle
Cyclic Quad	$\angle C = 180^\circ - \angle A$	$\angle A = 85^\circ$	$\angle C = 95^\circ$	Opposite angles sum to 180°
Intersecting Chords	$AP \cdot PB = CP \cdot PD$	$6, 4, 8$	$PD = 3.0$	Internal segment products equal
Tangent-Secant	$PT = \sqrt{PA \cdot PB}$	$PA=4, PB=16$	$PT = 8.0$	Power of an external point
Outside Arc Angle	$\theta = (\text{Arc}_1 - \text{Arc}_2)/2$	$110^\circ, 40^\circ$	$\theta = 35^\circ$	Secant intersection exterior angle

Real-World Engineering, Navigation & Robotics Applications

Everyday Engineering Implementations:

Maritime Navigation: Danger circles and horizontal sextant angle positioning rely on the Inscribed Angle Theorem to maintain safe clearance from underwater reefs.
Civil Architecture & Bridge Design: Calculating arch curvature, chord stress distributions, and circular pier load distributions using cyclic quadrilateral formulas.
Computer Graphics & Game Physics: Fast circular collision detection and ray-casting intersections use the Tangent-Secant power-of-a-point formula.
Robotics Control Systems: Gershgorin Circle Theorem bounds closed-loop system eigenvalues to guarantee dynamic stability without calculating full characteristic polynomials.

Frequently Asked Questions (FAQ)

Authoritative answers to common questions about circle theorems, inscribed angles, cyclic quadrilaterals, chord segments, and Gershgorin discs.

Theorem Name	Primary Formula	Sample Inputs	Calculated Result	Geometric Meaning
Inscribed Angle	\(\theta_i = \theta_c / 2\)	\(\theta_c = 120^\circ\)	\(\theta_i = 60^\circ\)	Center angle is 2x circumference
Thales Semicircle	\(\theta = 90^\circ\)	\(\theta_c = 180^\circ\)	\(\theta = 90^\circ\)	Right angle in semicircle
Cyclic Quad	\(\angle C = 180^\circ - \angle A\)	\(\angle A = 85^\circ\)	\(\angle C = 95^\circ\)	Opposite angles sum to 180°
Intersecting Chords	\(AP \cdot PB = CP \cdot PD\)	\(6, 4, 8\)	\(PD = 3.0\)	Internal segment products equal
Tangent-Secant	\(PT = \sqrt{PA \cdot PB}\)	\(PA=4, PB=16\)	\(PT = 8.0\)	Power of an external point
Outside Arc Angle	\(\theta = (\text{Arc}_1 - \text{Arc}_2)/2\)	\(110^\circ, 40^\circ\)	\(\theta = 35^\circ\)	Secant intersection exterior angle