medium difficulty5 min

Concyclic Points

Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary.

In one line: Four on one circle: opposite angles total 180.

Practice This Concept

Understand it

For three non-collinear points, a circle can always be drawn. For four points, the fourth point must also lie on that same circle. In Class 9 problems, opposite angles adding to 180 degrees is a common test.

Example

If in quadrilateral ABCD, angle A + angle C = 180 degrees, then A, B, C and D are concyclic.

Formula and rule

Concyclic test for four points: if one pair of opposite angles in the quadrilateral is supplementary, the four points lie on one circle.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1In quadrilateral PQRS, angle P = 75 degrees and angle R = 105 degrees.
2Since angle P + angle R = 180 degrees, points P, Q, R and S are concyclic.

Method you can reuse

1Name the quadrilateral using the four points.
2Find opposite angle pairs.
3If one pair adds to 180 degrees, write that the points are concyclic.

Visual tip

A diagram is useful but not always necessary; show four labelled points forming a quadrilateral and a circle through them.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If angle A + angle C = 180 degrees in quadrilateral ABCD, what does it show about A, B, C and D?

Think first, then show answer

It shows that A, B, C and D are concyclic because a quadrilateral with supplementary opposite angles is cyclic.

Common wrong answer

Adding adjacent angles and getting 180 degrees does not prove concyclicity; the test uses opposite angles.

Don't mix this up

Students may say any four points are concyclic. This is not true; the fourth point must satisfy the circle condition.

Exam tip

To prove four points concyclic, try to form a quadrilateral and check one pair of opposite angles.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions may ask students to prove four points lie on a circle using angle sums or cyclic quadrilateral converse.

1
Enough for 1 mark
Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary.
2
Add for 2 marks
Example: If in quadrilateral ABCD, angle A + angle C = 180 degrees, then A, B, C and D are concyclic.
3
Add for 3 marks
For three non-collinear points, a circle can always be drawn.

MCQ Quiz

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