medium difficulty8 min

Three Points Determine a Circle

One and only one circle can pass through three non-collinear points. Its centre is found at the intersection of perpendicular bisectors of two joining segments.

In one line: Three non-straight points fix one circle.

Practice This Concept

Understand it

Three points that do not lie on one straight line form a triangle. The perpendicular bisectors of its sides meet at one point, which is equidistant from all three points. That point becomes the centre of the required circle.

Example

For non-collinear points A, B and C, draw perpendicular bisectors of AB and BC. If they meet at O, then OA = OB = OC, so a circle with centre O passes through A, B and C.

Formula and rule

Unique circle condition: three points must be non-collinear.

Centre: intersection of perpendicular bisectors of two sides.

See it worked out

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

Worked example

Given three non-collinear points P, Q, R.

1Construct perpendicular bisectors of PQ and QR.
2Their intersection O gives OP = OQ = OR, so draw the circle with centre O and radius OP.

Method you can reuse

1Join the three points to form two segments.
2Construct perpendicular bisectors of two segments.
3Mark their intersection as centre.
4Draw circle using distance from centre to any one point.

Sketch it in your answer

Draw triangle ABC, perpendicular bisectors of AB and BC meeting at O, then draw circle through A, B and C.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why can three collinear points not determine a circle?

Think first, then show answer

Three collinear points cannot determine a circle because a circle would need one centre equidistant from all three, but the perpendicular bisectors do not meet in a suitable single centre for a straight-line set of three points.

Common wrong answer

Choosing the midpoint of only one side as centre.

Why it loses marks: It may be equidistant from two points, not necessarily from the third.

Don't mix this up

Students forget the condition non-collinear. Three points on a straight line cannot lie on one ordinary circle.

Exam tip

In construction or proof, mention that the centre is equidistant from all three points because it lies on perpendicular bisectors.

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

Exam questions may ask for construction of circumcircle of a triangle or proof that a circle through three non-collinear points is unique.

1
Enough for 1 mark
One and only one circle can pass through three non-collinear points. Its centre is found at the intersection of perpendicular bisectors of two joining segments.
2
Add for 2 marks
Example: For non-collinear points A, B and C, draw perpendicular bisectors of AB and BC. If they meet at O, then OA = OB = OC, so a circle with centre O passes through A, B and C.
3
Add for 3 marks
Three points that do not lie on one straight line form a triangle.

MCQ Quiz

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