MathematicsChapter 11: Practical Geometry of Circles
Circle Through Three Non-Collinear Points
Join A, B, C to make triangle ABC.
Learning objectives
- 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.
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 to A (or B or C). Collinear points: stop, no unique circle.
Where / symbols
- 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.
Overview
Triangle, then bisectors:
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.
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.
Examples Same circumcircle · Do not guess the radius · Arc is part of that unique circle
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.
Worked example
- 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
In simple words
Medians meet at the centroid, altitudes at the orthocentre. The circle needs the perp bisectors, not those.
Key points
- triangle first
- perp bisectors of sides
- circumcentre
- three points on the arc
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Circle Through Three Non-Collinear Points
Sticky Notes · Quick Revision
Core Idea
- 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.
Worked Method
- 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 Misconception
- Medians meet at the centroid, altitudes at the orthocentre. The circle needs the perp bisectors, not those.
FBISE Exam Check
- Medians meet at the centroid, altitudes at the orthocentre. The circle needs the perp bisectors, not those.
60-second revision
- 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
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