# How do you prove a cyclic quadrilateral is a circle?

## How do you prove a cyclic quadrilateral is a circle?

If all the four vertices of a quadrilateral ABCD lie on the circumference of the circle, then ABCD is a cyclic quadrilateral. In other words, if any four points on the circumference of a circle are joined, they form the vertices of a cyclic quadrilateral.

## How do you prove the circle theorem?

Draw two radii as shown. Since an angle subtended at the circumference by an arc is half that subtended at the centre, the angles round the centre are 2a and 2b. Angles round a point add up to 360° so 2a + 2b = 360°. Therefore a + b = 180°, so the theorem is proven.

What is cyclic quadrilateral in circle?

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.

### What is circle theorem?

Circle theorem may refer to: Any of many theorems related to the circle; often taught as a group in GCSE mathematics. These include: Inscribed angle theorem. Thales’ theorem, if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ∠ABC is a right angle.

### What are theorems and proofs?

proofA proof is a series of true statements leading to the acceptance of truth of a more complex statement. is the hypotenuse of the triangle. theoremA theorem is a statement that can be proven true using postulates, definitions, and other theorems that have already been proven.

How do you prove a quadrilateral theorem is inscribed?

Proof: In the quadrilateral ABCD can be inscribed in a circle, then we have seen above using the inscribed angle theorem that the sum of either pair of opposite angles = (1/2(a1 + a2 + a3 + a4) = (1/2)360 = 180. Conversely, if the quadrilateral cannot be inscribed, this means that D is not on the circumcircle of ABC.

## How do you prove a cyclic quadrilateral is a rectangle?

Each angle of a rectangle is a right angle. For a cyclic quadrilateral, sum of opposite angles is 180°. => 90° + 90° = 180° ( sum of opposite angles of a rectangle ). Hence, rectangle is a cyclic quadrilateral.

## What are the circle theorem rules?

Circle Theorems

• Theorem 1: The angle in a semicircle is 90°
• Theorem 2: The angle at the centre is double the angle at the circumference.
• Theorem 3: Angles from the same chord in the same segment are equal.
• Theorem 4: Opposite angles in a cyclic quadrilateral sum to 180 °
• Theorem 5: Alternate segment theorem.

How do you prove the diameter of a circle?

1. Given : In circle with center O,CD is the diameter and AB is the chord which is bisected by diameter at E,OA and OB are joined.
2. To prove : ∠AOE=∠BOE.
3. Proof: In ΔOAE and ΔOBE.
4. OA=OB (Radii of the circle)
5. OE=OE (Common)
6. AE=BE (Given)
7. ∴ ΔOAE≅ΔOBE (By SSS congruency criterian)
8. ∴ ∠AOE=∠BOE (CPCT)

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