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.
Practice This ConceptMain explanation
Teacher explanation
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.
Simple analogy
Three non-straight points fix one circle.
Common confusion
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.
Answer writing and exam use
1-mark use
Write the exact meaning of three points determine a circle in one clean line.
2-mark use
Define three points determine a circle and add one example or condition.
3-mark use
Explain three points determine a circle, show the method or example, and mention the common mistake.
Practice this concept with focused MCQs
Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.
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