Four on a circle
Concyclic points and cyclic quadrilaterals
A fourth point
Three points that are not in a straight line always have a circle through them. Now a fourth point D joins A, B and C.
A, B and C have a circle through them. You place D somewhere on the page. Will D be on that circle?
What this lesson covers
The idea
Points that lie on the same circle are called concyclic; a quadrilateral (4-gon) whose vertices are concyclic is called a cyclic quadrilateral.
A fourth point
Three points that are not in a straight line always have a circle through them. Now a fourth point D joins A, B and C.
A, B and C have a circle through them. You place D somewhere on the page. Will D be on that circle?
- Always
- Only in some places
- Never
Place D
A, B and C are fixed, and the dashed circle goes through them. Touch the shaded region to place D.
On one circle
Points that lie on the same circle are called concyclic. A quadrilateral whose four vertices are concyclic is called a cyclic quadrilateral.
The circle through A, B and C is fixed. D is on it only in exactly the right places. A point D inside or outside the circle is not concyclic with A, B, C. When D is on the circle, ABCD is a cyclic quadrilateral.
Notes
Points on the same circle are concyclic. A quadrilateral with its four vertices on a circle is a cyclic quadrilateral.
Check yourself
Which word describes points that lie on the same circle?
A, B and C are three points that are not on one straight line. How many circles pass through all three of them?
Answer: 1
Three points not in a line have exactly 1 circle through them, the circumcircle of triangle ABC.
A circle through A, B and C has radius 5 cm and centre O. Which point D is concyclic with A, B and C?
The four vertices of a quadrilateral ABCD lie on one circle. What do we call ABCD?
- Collinear. Collinear points lie on the same straight line. Points on the same circle are called concyclic.
- Congruent. Congruent figures are exactly equal in shape and size. Points on the same circle are called concyclic.
- Concyclic — correct. Yes. Points on the same circle are concyclic.
- D with OD = 5 cm — correct. Yes. A point 5 cm from O is on the circle, so A, B, C, D are concyclic.
- D with OD = 4 cm. A point 4 cm from O is inside the circle, so it is not on it.
- D with OD = 6 cm. A point 6 cm from O is outside the circle, so it is not on it.
- A parallelogram. Having four vertices on a circle does not make the opposite sides parallel. It is called a cyclic quadrilateral.
- A cyclic quadrilateral — correct. Yes. The points are concyclic, and the quadrilateral is a cyclic quadrilateral.
- A collinear quadrilateral. Collinear means on one line. A quadrilateral with its vertices on a circle is called cyclic.