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Circle Through Three Non-collinear Points
Join A, B, C to make triangle ABC. The circumcentre is where the perpendicular bisectors of the sides meet. That point is the centre; the radius is the distance to A (or B or C). Collinear points: stop, no unique circle.
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Circle Through Three Non-collinear Points
What is it?
Join A, B, C to make triangle ABC. The circumcentre is where the perpendicular bisectors of the sides meet. That point is the centre; the radius is the distance to A (or B or C). Collinear points: stop, no unique circle.
In simple words
- Join A, B, C to make triangle ABC.
- THE CONSTRUCTION: Draw sides.
- COMPLETING AN ARC: To complete a circle from a given arc, pick three points on the arc and use the same circumcentre.
- Worked-method reminder: open compass OA and draw circle centre O
Important terms
CIRCLE THROUGH THREE NON-COLLINEAR POINTS: Join A, B, C to make triangle ABC. The circumcentre is where the perpendicular bisectors of the sides meet. That point is the centre; the radius is the distance to A (or B or C). Collinear points: stop, no unique circle.
THE CONSTRUCTION: Draw sides. Perp bisect two of them (the third is a check). Compass open to OA and draw the circle. It will pass through B and C by construction.
COMPLETING AN ARC: To complete a circle from a given arc, pick three points on the arc and use the same circumcentre. You are finding the original centre.
What you should learn
- Explain circle through three non-collinear points using the correct definitions, conditions, and notation.
- Apply triangle first, perp bisectors of sides correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- Join ABC
- Perp bisectors meet
- Radius OA
THE CONSTRUCTION
Draw sides. Perp bisect two of them (the third is a check). Compass open to OA and draw the circle. It will pass through B and C by construction.
COMPLETING AN ARC
To complete a circle from a given arc, pick three points on the arc and use the same circumcentre. You are finding the original centre.
Recommended for Class 10
Full school-level explanation
Join A, B, C to make triangle ABC. The circumcentre is where the perpendicular bisectors of the sides meet. That point is the centre; the radius is the distance to A (or B or C). Collinear points: stop, no unique circle.
Triangle, then bisectors:
WHAT — CIRCLE THROUGH THREE NON-COLLINEAR POINTS
Join A, B, C to make triangle ABC. The circumcentre is where the perpendicular bisectors of the sides meet. That point is the centre; the radius is the distance to A (or B or C). Collinear points: stop, no unique circle. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Circle Through Three Non-Collinear Points matters because accurate compass-and-straightedge constructions apply circle theorems in a visible way. Each construction works because a locus or perpendicular condition fixes the required centre, circle, or tangent.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through THE CONSTRUCTION, then COMPLETING AN ARC. Worked: draw the circle through A, B, C (named steps). check A, B, C are not collinear draw triangle ABC construct perp bisector of AB construct perp bisector of BC label intersection O open compass OA and draw circle centre O Finish by checking signs, restrictions, units, and whether the answer matches the question.
THE CONSTRUCTION
Draw sides. Perp bisect two of them (the third is a check). Compass open to OA and draw the circle. It will pass through B and C by construction.
Remember
- circumcentre
Examples
- Join ABC
- Perp bisectors meet
- Radius OA
COMPLETING AN ARC
To complete a circle from a given arc, pick three points on the arc and use the same circumcentre. You are finding the original centre.
Remember
- three points on the arc
Examples
- Same circumcircle
- Do not guess the radius
- Arc is part of that unique circle
Remember
- triangle first
- perp bisectors of sides
- circumcircle of ABC
- not medians
- circumcentre
- three points on the arc
- triangle first
- perp bisectors of sides
How the ideas connect
Worked: draw the circle through A, B, C (named steps). check A, B, C are not collinear draw triangle ABC construct perp bisector of AB construct perp bisector of BC label intersection O open compass OA and draw circle centre O
Deeper understanding
Go beyond memorising Circle Through Three Non-collinear Points. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: draw the circle through A, B, C (named steps). check A, B, C are not collinear draw triangle ABC construct perp bisector of AB construct perp bisector of BC label intersection O open compass OA and draw circle centre O
Common mistakes
Exam focus
Core Idea
THE CONSTRUCTION
COMPLETING AN ARC
Worked Method
Exam focus
Quick revision
- triangle first
- perp bisectors of sides
- circumcircle of ABC
- not medians
- CIRCLE THROUGH THREE NON-COLLINEAR POINTS: Join A, B, C to make triangle ABC. The circumcentre is where the perpendicular bisectors of the side
- THE CONSTRUCTION: Draw sides. Perp bisect two of them (the third is a check). Compass open to OA and draw the circle.
Medians meet at the centroid, altitudes at the orthocentre. The circle needs the perp bisectors, not those.
For long answers on Circle Through Three Non-collinear Points: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- triangle first
- perp bisectors of sides
- circumcircle of ABC
- not medians
Important formulas
- triangle first
- perp bisectors of sides
- circumcircle of ABC
Common exam traps
- Medians meet at the centroid, altitudes at the orthocentre. The circle needs the perp bisectors, not those.
- Do not guess the radius
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Medians meet at the centroid, altitudes at the orthocentre. The circle needs the perp bisectors, not those.
- Mistake
Do not guess the radius
- Mistake
Do not treat THE CONSTRUCTION and COMPLETING AN ARC as the same step; each has a different purpose.
- Mistake
Medians meet at the centroid, altitudes at the orthocentre. The circle needs the perp bisectors, not those.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
triangle first
perp bisectors of sides
circumcircle of ABC
not medians
Join A, B, C to make triangle ABC. The circumcentre is where the perpendicular bisectors of the sides meet. That point is the centre; the radius is the distance
- Definition
- Join A, B, C to make triangle ABC. The circumcentre is where the perpendicular bisectors of the sides meet. That point is the centre; the radius is the distance
- Example
- Join ABC
- Critical warning
- Medians meet at the centroid, altitudes at the orthocentre. The circle needs the perp bisectors, not those.
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Exam Practice
Define
Q1. Define Circle Through Three Non-collinear Points.
Explain
Q2. Explain: THE CONSTRUCTION.
Example
Q3. Give one clear example of Circle Through Three Non-collinear Points and say why it fits.
Mistake
Q4. State one common mistake students make about Circle Through Three Non-collinear Points.
Exam note
Q5. Write a short exam note on Circle Through Three Non-collinear Points: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.