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Cyclic and Tangential Quadrilaterals

A cyclic quadrilateral has all four corners on one circle. Its opposite angles add up to 180°, and each outside angle equals the opposite inside angle. A tangential quadrilateral has a circle inside touching all four sides; there, opposite sides have equal sums: AB + CD = BC + DA. The converse of the intercept (Thales) theorem tells when a line is parallel to a side of a triangle.

🎬 Step-by-step story

  1. Here is a circle with four dots on its edge. Call them A, B, C and D.
  2. Join the dots in order. This four-sided shape is a cyclic quadrilateral. "Cyclic" means all four corners sit on one circle.
  3. Look at two opposite corners, A and C. Their angles add up to 180°. The other pair, B and D, also adds up to 180°.
  4. Slide point D along the circle. Every angle changes, but each opposite pair still adds up to 180°.
  5. Stretch one side outwards. The angle outside the shape is equal to the inside angle at the opposite corner.
  6. Now the circle sits inside and touches all four sides. Here opposite sides give the same total: AB + CD = BC + DA. Move the touch point and check.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why do the opposite angles add to 180° and not some other number?

Two centre angles make a full turn of 360°, and each edge angle is half of its centre angle. Half of 360° is 180°.

If I move one corner, is the rule still true?

Yes, as long as the corner stays on the circle. Slide D and watch: the angles change, the sums do not.

What is the outside angle rule for?

It saves time. When a side is extended, you can read the opposite inside angle directly, with no subtraction.

Is a rectangle cyclic?

Yes. Its opposite angles are both 90°, so they add to 180°. All four corners fit on a circle whose centre is the middle of the rectangle.

Can a shape have both a circle through its corners and a circle inside?

Yes, a square can. Then both rules hold together.

Why are opposite sides equal in sum when a circle is inside?

From each corner the two tangent lengths are equal. When you add around the shape, each tangent length appears once on each side of the equation.

Quadrilaterals inscribed in a circle

A cyclic quadrilateral is a four-sided shape whose four corners all lie on one circle. We say the quadrilateral is inscribed in the circle. Four points on a circle are called concyclic.

Main rule: opposite angles add up to 180°. So ∠A + ∠C = 180° and ∠B + ∠D = 180°.

Outside angle rule: if you extend one side, the angle outside at that corner equals the inside angle at the opposite corner.

Converse: if a quadrilateral has one pair of opposite angles adding to 180°, then it is cyclic.

Why opposite angles add up to 180°

Take the centre O. The angle at the centre on arc BCD is twice ∠A (angle at the edge). The angle at the centre on the other arc BAD is twice ∠C. The two centre angles together make one full turn, 360°. So 2∠A + 2∠C = 360°. Divide by 2: ∠A + ∠C = 180°.

Which quadrilaterals can be cyclic? Squares, rectangles and isosceles trapeziums always can. A parallelogram is cyclic only if it is a rectangle, because its opposite angles are equal and must also add to 180°, so each is 90°. A rhombus that is not a square is never cyclic.

Quadrilaterals around a circle (tangential)

A tangential quadrilateral has a circle inside it that touches all four sides. Every side is a tangent. From one corner, the two tangents to the circle are equal in length. Adding these equal pairs round the shape gives:

AB + CD = BC + DA (sums of opposite sides are equal). This is the Pitot theorem.

The converse is also true: if the sums of opposite sides are equal, a circle fits inside. Squares, rhombuses and kites have an incircle. For any shape with an incircle, Area = r × s where r is the inradius and s is half the perimeter.

A shape can be both: a square is cyclic and tangential.

Converse of the intercept (Thales) theorem

The intercept theorem says: a line parallel to one side of a triangle cuts the other two sides in the same ratio. The converse says: if a line cuts two sides of a triangle in the same ratio, the line is parallel to the third side.

In triangle ABC, let D be on AB and E on AC. If AD/DB = AE/EC, then DE ∥ BC. This is a quick way to prove two lines are parallel, and it often helps to show that a quadrilateral is a trapezium before checking whether it is cyclic.

Typical exam questions

Try it at home

Draw a circle with a bowl or a coin. Mark four points on it and join them. Measure opposite angles with a protractor and add them. Do you get 180°, give or take a degree or two? Now cut a kite shape from paper and try to fit a coin inside, touching all four sides. Then check the side sums.

Key formulas and definitions

Worked examples

1. ABCD is cyclic and ∠A = 75°. Find ∠C.

Opposite angles add to 180°. ∠C = 180° − 75° = 105°.

2. ABCD is cyclic and ∠B = 110°. Find ∠D.

∠D = 180° − 110° = 70°.

3. In a cyclic quadrilateral the side AB is extended to P. The outside angle at B is 85°. Find the inside angle at D.

The outside angle at B equals the opposite inside angle ∠D. So ∠D = 85°. (Check: ∠B = 180° − 85° = 95°, and 95° + 85° = 180°.)

4. The angles A, B, C of a cyclic quadrilateral are in the ratio 2 : 3 : 4. Find all four angles.

∠A + ∠C = 180°, so 2x + 4x = 180°, x = 30°. ∠A = 60°, ∠B = 90°, ∠C = 120°. ∠D = 180° − 90° = 90°.

5. A quadrilateral has sides AB = 7, BC = 6, CD = 9. A circle touches all four sides. Find DA.

AB + CD = 7 + 9 = 16. This equals BC + DA = 6 + DA. So DA = 10.

6. A tangential quadrilateral has perimeter 36 cm and inradius 4 cm. Find its area.

s = 36 ÷ 2 = 18 cm. Area = r × s = 4 × 18 = 72 cm².

7. Prove: a parallelogram inscribed in a circle is a rectangle.

In a parallelogram opposite angles are equal: ∠A = ∠C. In a cyclic quadrilateral ∠A + ∠C = 180°. So 2∠A = 180°, ∠A = 90°. A parallelogram with one right angle is a rectangle.

8. In triangle ABC, D is on AB with AD = 3, DB = 6, and E is on AC with AE = 2, EC = 4. Is DE parallel to BC?

AD/DB = 3/6 = 1/2 and AE/EC = 2/4 = 1/2. The ratios are equal, so by the converse of the intercept theorem, DE ∥ BC.

Common mistakes

Practice quiz

1. In a cyclic quadrilateral, opposite angles add up to:
2. Which quadrilateral is NOT always cyclic?
3. For a quadrilateral with an incircle, which is true?
4. The outside angle at one corner of a cyclic quadrilateral equals:
5. If AD/DB = AE/EC in triangle ABC, then:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is a cyclic quadrilateral?

A four-sided shape whose four corners all lie on one circle. Its opposite angles add to 180°.

How do I prove four points are concyclic?

Show that one pair of opposite angles of the quadrilateral adds up to 180°, or that an outside angle equals the opposite inside angle.

What is the difference between cyclic and tangential?

In a cyclic quadrilateral the circle passes through the four corners. In a tangential quadrilateral the circle sits inside and touches the four sides.

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