I’m Up and Down, and Round and Round · Lesson 7 of 8
Concyclicity and Cyclic Quadrilaterals
“Put four points on one circle and their angles start sharing secrets.”
• Define concyclic points and cyclic quadrilaterals. • Use equal subtended angles to prove concyclicity. • Prove that opposite angles of a cyclic quadrilateral sum to 180°. • Understand the converse theorem. • Apply cyclic-quadrilateral properties to angle problems.
Concyclicity of Points
Points that lie on the same circle are called concyclic.
Three non-collinear points always determine a circle. With four points, we need an additional condition to know whether the fourth lies on that same circle.
If segment AB subtends equal angles at two points C and D on the same side of AB, then A, B, C and D are concyclic.
Start with the unique circle through A, B and C. If D were inside or outside this circle, angle comparisons using exterior angles would force a contradiction with the given equality ∠ACB=∠ADB. Therefore D must lie on the circle.
Cyclic Quadrilateral
A quadrilateral whose four vertices lie on a single circle is called a cyclic quadrilateral.
Opposite Angles Are Supplementary
The sum of each pair of opposite angles of a cyclic quadrilateral is 180°.
In cyclic quadrilateral ABCD, angle BAD is half the central angle subtended by arc BCD. Angle BCD is half the central angle subtended by the other arc BAD. Those two central angles together make a complete 360° turn. Therefore the two opposite angles add to half of 360°, which is 180°.
Problem
In cyclic quadrilateral ABCD, ∠A=75° and ∠B=110°. Find ∠C and ∠D.
- 1.Opposite angles are supplementary.
- 2.∠C=180°−75°=105°.
- 3.∠D=180°−110°=70°.
The Converse
If a pair of opposite angles of a quadrilateral adds to 180°, then the four vertices are concyclic.
This converse is powerful because it allows us to prove that a quadrilateral is cyclic without drawing its circle first. If opposite angles are supplementary, a circle through three vertices must also pass through the fourth.
Problem
A quadrilateral has angles 80°, 110°, 100° and 70° in order. Can it be cyclic?
- 1.First opposite pair: 80°+100°=180°.
- 2.Second opposite pair: 110°+70°=180°.
- 3.Therefore the quadrilateral satisfies the cyclic condition.
- 4.So such a cyclic quadrilateral can be drawn.
Exterior Angle Connection
Because opposite interior angles of a cyclic quadrilateral sum to 180°, an exterior angle at one vertex equals the interior opposite angle. The exterior angle and its adjacent interior angle also sum to 180°, so both are supplements of the same opposite angle.
Practice Problems
- In a cyclic quadrilateral, ∠A=68°. Find ∠C.
- If ∠P=(2x+10)° and opposite angle ∠R=(3x−20)°, find x and both angles.
- A quadrilateral has opposite angles 92° and 88°. What can you conclude?
- Explain why equal angles subtended by the same segment can be used to prove four points are concyclic.
- Prove that the exterior angle of a cyclic quadrilateral equals the interior opposite angle.
- Show that a rectangle can be inscribed in a circle by using its opposite angles.
Key Takeaways
• Concyclic points lie on one circle. • Equal angles subtended by the same segment can establish concyclicity. • A cyclic quadrilateral has supplementary opposite angles. • The converse also holds: supplementary opposite angles imply cyclicity. • Exterior angle of a cyclic quadrilateral equals the interior opposite angle.
Next, we bring together all the circle theorems and practise choosing the right result for each problem.