Cyclic Quadrilaterals

A cyclic quadrilateral is a quadrilateral whose four vertices all lie on a single circle, called the circumscribed circle or circumcircle. Not every quadrilateral can be inscribed in a circle — only those satisfying special angle relationships qualify. Cyclic quadrilaterals appear throughout classical geometry and have elegant properties that make them powerful tools for solving problems involving circles, angles, and lengths. The most fundamental property of a cyclic quadrilateral is that its opposite angles are supplementary — they add up to 180°. This single theorem unlocks a remarkable chain of deductions: you can find unknown angles, prove that four points are concyclic, and connect cyclic quadrilaterals to arc measures and the inscribed angle theorem. These ideas form the backbone of Euclidean circle geometry at the high school level. Cyclic quadrilaterals have been studied since antiquity. Ptolemy of Alexandria used them to derive his famous theorem relating the diagonals and sides of a cyclic quadrilateral, a result that encodes the sine addition formula in purely geometric language. Understanding cyclic quadrilaterals sharpens your ability to reason deductively — constructing logical chains from given information to proven conclusions — which is the essence of Euclidean geometry.

Key takeaways
  • Definition and the Circumcircle
  • Opposite Angles are Supplementary
  • Converse: Test for a Cyclic Quadrilateral
  • Exterior Angle Property
  • Ptolemy's Theorem
  • Special Cases

Definition and the Circumcircle

A quadrilateral ABCD is cyclic if and only if all four vertices A, B, C, D lie on a common circle. Every triangle has a circumcircle, but for a quadrilateral this is a special condition. The circle is called the circumscribed circle, and the quadrilateral is said to be inscribed in it.

Opposite Angles are Supplementary

The central theorem: in cyclic quadrilateral ABCD, opposite angle pairs sum to 180°. That is, ∠DAB + ∠BCD = 180° and ∠ABC + ∠CDA = 180°. Proof sketch: ∠DAB is an inscribed angle subtending arc BCD, so ∠DAB = ½·arc(BCD). Similarly ∠BCD = ½·arc(BAD). Since arc(BCD) + arc(BAD) = 360° (the full circle), we get ∠DAB + ∠BCD = ½·360° = 180°. This is a direct consequence of the Inscribed Angle Theorem.

Converse: Test for a Cyclic Quadrilateral

The converse is equally important for deductive proofs: if the opposite angles of a quadrilateral sum to 180°, then the quadrilateral is cyclic. This gives you a way to prove four points are concyclic without explicitly finding the circle. If you can show ∠A + ∠C = 180° (or equivalently ∠B + ∠D = 180°), the quadrilateral must be inscribed in some circle.

Exterior Angle Property

An exterior angle of a cyclic quadrilateral equals the interior opposite angle. If you extend side CD beyond D to a point E, then ∠CDE (the exterior angle at D) equals ∠ABC (the interior angle at B). This follows directly from the supplementary angle property: ∠CDA + ∠ABC = 180°, and ∠CDE + ∠CDA = 180° (angles on a straight line), so ∠CDE = ∠ABC.

Ptolemy's Theorem

For a cyclic quadrilateral ABCD with sides a = AB, b = BC, c = CD, d = DA and diagonals p = AC, q = BD, Ptolemy's Theorem states: p·q = a·c + b·d. In words, the product of the diagonals equals the sum of the products of opposite sides. This theorem is both a necessary and sufficient condition for a quadrilateral to be cyclic. It can be used to derive the sine addition and subtraction formulas, showing the deep connection between circle geometry and trigonometry.

Special Cases

Several familiar shapes are cyclic quadrilaterals. A rectangle is always cyclic because all its angles are 90°, and opposite angles trivially sum to 180°. An isosceles trapezoid is also always cyclic. A square is cyclic. A general parallelogram is cyclic only if it is a rectangle, because in a parallelogram opposite angles are equal, and equal angles can only both be 90° if they are to sum to 180°.

The Key Equation

∠DAB+∠BCD=180∘and∠ABC+∠CDA=180∘\angle DAB + \angle BCD = 180^\circ \quad \text{and} \quad \angle ABC + \angle CDA = 180^\circ
  • \angle DAB — Angle at vertex A (degrees): Interior angle of the quadrilateral at vertex A, subtending arc BCD as an inscribed angle.
  • \angle BCD — Angle at vertex C (degrees): Interior angle at vertex C, opposite to angle A; subtends arc BAD. Together with ∠DAB it spans the full 360° of the circle.
  • 180^\circ — Supplementary sum (degrees): The target sum for supplementary angles — equal to half of the 360° total arc of the circumscribed circle.

Worked Example

Problem. Quadrilateral ABCD is inscribed in a circle. Angle A = 3x + 10° and angle C = 5x − 30°. Find the value of x and the measures of all four angles, given that angle B = 2y + 15° and angle D = y + 45°.

x=25,  y=40;∠A=85∘,  ∠B=95∘,  ∠C=95∘,  ∠D=85∘x = 25,\; y = 40;\quad \angle A = 85^\circ,\; \angle B = 95^\circ,\; \angle C = 95^\circ,\; \angle D = 85^\circ

Answer. x = 25, y = 40; ∠A = 85°, ∠B = 95°, ∠C = 95°, ∠D = 85°

Figure
Diagram
ABCDO∠A∠B∠C∠D∠A + ∠C = 180° and ∠B + ∠D = 180°p = ACq = BD
Origins · 1 minClaudius Ptolemy and how this was discovered
CP
Claudius Ptolemy
100–170 · Greco-Egyptian (Roman Alexandria)

Ptolemy proved the theorem bearing his name in his landmark astronomical treatise the Almagest (circa 150 CE): for a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides (AC · BD = AB · CD + AD · BC). He used this result to construct his table of chords — the ancient equivalent of a trigonometric table — which powered Greek astronomy for over a millennium. Ptolemy's theorem is essentially a disguised version of the sine addition formula. By choosing a specific cyclic quadrilateral inscribed in a unit circle, the theorem instantly yields sin(α + β) = sin α cos β + cos α sin β, connecting ancient geometry to modern trigonometry in a single elegant step.

  • Inscribed Angle Theorem — The opposite-angles property of cyclic quadrilaterals is a direct corollary of the Inscribed Angle Theorem, which states that an inscribed angle is half the central angle subtending the same arc.
  • Circle Theorems — Cyclic quadrilateral properties are one of several important circle theorems studied in Euclidean geometry, alongside tangent-radius relationships and angles in semicircles.
  • Ptolemy's Theorem — Ptolemy's Theorem gives a metric relationship between the sides and diagonals of a cyclic quadrilateral, extending the angle properties to include lengths.
  • Deductive Reasoning in Geometry — Proving that a quadrilateral is cyclic using angle sums is a classic exercise in constructing a logical deductive argument from given hypotheses to a geometric conclusion.