Mathematics.

circles

Circle Theorems

Geometry35 minDifficulty5 out of 10

You should know: circles

Overview

Circle theorems are a family of results describing the relationships between angles, chords, tangents, and arcs of a circle. They build directly on the inscribed angle theorem — that an inscribed angle is always half the central angle subtending the same arc — and extend it to cyclic quadrilaterals, tangent lines, and intersecting chords. These theorems let you deduce unknown angles and lengths in circle diagrams without any measurement, using only the circle's defining property that every radius has the same length.

Intuition

Picture a central angle as the 'true' measurement of an arc, since it's measured directly from the circle's center. Any inscribed angle looking at that same arc from a point on the circle sees only half of that true measurement — no matter where on the remaining arc the vertex sits, because moving the vertex around the circle keeps the subtended arc, and hence the central angle, unchanged. This single fact cascades into everything else: a cyclic quadrilateral's opposite angles are supplementary because each pair of opposite angles subtends arcs summing to the whole circle (360°) and each angle is half its arc; the tangent-radius perpendicularity follows because a tangent line is the limiting case of a chord as its two intersection points merge into one.

Interactive Graph

Drag a point around a circle and watch the inscribed angle stay fixed at half the central angle

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Formal Definition

Definition

For a circle with center O, an inscribed angle θᵢ subtending an arc with central angle θ꜀ at the same arc, and a cyclic quadrilateral with opposite angles α, γ and β, δ:

θi=12θc\theta_i = \tfrac{1}{2}\theta_c
Inscribed angle theorem
θi=90∘ when the subtended arc is a semicircle\theta_i = 90^\circ \text{ when the subtended arc is a semicircle}
Angle in a semicircle (Thales' theorem)
α+γ=180∘,β+δ=180∘\alpha + \gamma = 180^\circ, \qquad \beta+\delta = 180^\circ
Opposite angles of a cyclic quadrilateral are supplementary
OT⊥ℓ at the point of tangency TOT \perp \ell \text{ at the point of tangency } T
Tangent–radius theorem: a tangent line is perpendicular to the radius at the point of tangency

Notation

NotationMeaning
OOThe center of the circle
TTPoint of tangency, where a tangent line touches the circle
θi, θc\theta_i,\ \theta_cInscribed angle and its corresponding central angle on the same arc

Properties

Inscribed angle theorem

θi=12θc\theta_i = \tfrac{1}{2}\theta_c

Condition: The inscribed angle and central angle must subtend the same arc.

Example: A central angle of 80° gives an inscribed angle of 40° on the same arc.

Angle in a semicircle (Thales' theorem)

Any inscribed angle subtending a diameter is exactly 90∘.\text{Any inscribed angle subtending a diameter is exactly } 90^\circ.

Inscribed angles on the same arc are equal

All inscribed angles subtending the same arc from the same side are congruent.\text{All inscribed angles subtending the same arc from the same side are congruent.}

Cyclic quadrilateral opposite angles

α+γ=180∘\alpha + \gamma = 180^\circ

Condition: Holds for any quadrilateral whose four vertices all lie on a single circle.

Tangent–radius perpendicularity

A tangent line is perpendicular to the radius drawn to the point of tangency.\text{A tangent line is perpendicular to the radius drawn to the point of tangency.}

Two tangents from an external point

The two tangent segments from a common external point to a circle are equal in length.\text{The two tangent segments from a common external point to a circle are equal in length.}

Applications

Tangent-radius perpendicularity is used to design cams, gears, and belt-and-pulley systems where a straight component must meet a circular one cleanly.

Worked Examples

  1. Apply the inscribed angle theorem: the inscribed angle is half the central angle.

    θi=12(100∘)=50∘\theta_i = \tfrac{1}{2}(100^\circ) = 50^\circ

Answer: 50°

Practice Problems

Difficulty 3/10

An inscribed angle subtends an arc with central angle 130°. Find the inscribed angle.

Difficulty 3/10

A triangle is inscribed in a circle so that one of its sides is a diameter. If one of the other two angles is 35°, find the third angle.

Difficulty 6/10

A cyclic quadrilateral ABCD has ∠A = 2x + 10° and ∠C = 3x − 30°. Find x and both angle measures.

Common Mistakes

Common Mistake

Applying the inscribed angle theorem to two angles that subtend different arcs.

The 'half the central angle' relationship only holds when the inscribed and central angles subtend the SAME arc — always check which arc is being subtended before comparing angles.

Common Mistake

Assuming any quadrilateral has supplementary opposite angles.

That property holds only for cyclic quadrilaterals — ones whose four vertices lie on a common circle. A general quadrilateral's opposite angles need not be supplementary.

Quiz

An inscribed angle subtending a diameter measures:
In a cyclic quadrilateral, opposite angles always:
Two tangent segments drawn from the same external point to a circle are:

Summary

  • Inscribed angle theorem: an inscribed angle is half the central angle subtending the same arc.
  • Thales' theorem: an angle inscribed in a semicircle (subtending a diameter) is always 90°.
  • All inscribed angles subtending the same arc from the same side are equal.
  • Opposite angles of a cyclic quadrilateral are supplementary (sum to 180°).
  • A tangent line is perpendicular to the radius at the point of tangency, and two tangents from one external point are equal in length.

References

  1. WebsiteWikipedia — Inscribed angle
  2. WebsiteWikipedia — Circle