MathematicsChapter 9: Chord and Arcs of a Circle
Unique Circle Through Three Points
Three non-collinear points lie on exactly one circle (their circumcircle).
Learning objectives
- Explain unique circle through three points using the correct definitions, conditions, and notation.
- Apply 3 non-collinear -> unique circle correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
Three non-collinear points lie on exactly one circle (their circumcircle). Three collinear points do not determine a circle. The centre is the meeting of the perpendicular bisectors of the chords joining the points (also Chapter 11's construction).
Mathematical form
AB = points equidistant from A and B perp bis
BC = points equidistant from B and C their in
OA = OB
Where / symbols
- UNIQUE CIRCLE THROUGH THREE POINTS
- Three non-collinear points lie on exactly one circle (their circumcircle). Three collinear points do not determine a circle. The centre is the meeting of the perpendicular bisectors of the chords joining the points (also Chapter 11's construction).
- UNIQUE CIRCUMCIRCLE
- A, B, C not on a line: one circle. Its centre is equidistant from A, B, and C (the circumradius). That distance is the radius.
- COLLINEAR FAIL
- Three on a line have no finite circumcentre (perp bisectors are parallel or the same). 'Through any three points' is false without 'non-collinear'.
Overview
Collinear or not:
Unique Circumcircle
A, B, C not on a line: one circle. Its centre is equidistant from A, B, and C (the circumradius). That distance is the radius.
Examples Triangle ABC · Perp bisectors meet · Not infinitely many
Collinear Fail
Three on a line have no finite circumcentre (perp bisectors are parallel or the same). 'Through any three points' is false without 'non-collinear'.
Examples Three on a line: no unique circle · Two points: infinitely many · One point: even more
WHAT — UNIQUE CIRCLE THROUGH THREE POINTS
Three non-collinear points lie on exactly one circle (their circumcircle). Three collinear points do not determine a circle. The centre is the meeting of the perpendicular bisectors of the chords joining the points (also Chapter 11's construction). The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Unique Circle Through Three Points matters because chord and arc theorems convert circle diagrams into equal lengths, equal angles, and right triangles. They are the foundation for later tangent proofs and constructions.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through UNIQUE CIRCUMCIRCLE, then COLLINEAR FAIL. Worked: why do perpendicular bisectors of AB and BC give the centre? perp bisector of AB = points equidistant from A and B perp bisector of BC = points equidistant from B and C their intersection O satisfies OA = OB = OC so O is the unique circumcentre Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- why do perpendicular bisectors of AB and BC give the centre?
- perp bisector of AB = points equidistant from A and B
- perp bisector of BC = points equidistant from B and C
- their intersection O satisfies OA = OB = OC
- so O is the unique circumcentre
In simple words
Two points determine infinitely many circles (centres on the perp bisector). You need the third point to pin the centre.
Key points
- 3 non-collinear -> unique circle
- equidistant centre
- need a triangle
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Unique Circle Through Three Points
Sticky Notes · Quick Revision
Core Idea
- Three non-collinear points lie on exactly one circle (their circumcircle). Three collinear points do not determine a circle. The centre is the meeting of the perpendicular bisectors of the chords joining the points (also Chapter 11's construction).
UNIQUE CIRCUMCIRCLE
- A, B, C not on a line: one circle. Its centre is equidistant from A, B, and C (the circumradius). That distance is the radius.
COLLINEAR FAIL
- Three on a line have no finite circumcentre (perp bisectors are parallel or the same). 'Through any three points' is false without 'non-collinear'.
Worked Method
- Worked: why do perpendicular bisectors of AB and BC give the centre?
- perp bisector of AB = points equidistant from A and B
- perp bisector of BC = points equidistant from B and C
- their intersection O satisfies OA = OB = OC
- so O is the unique circumcentre
Common Misconception
- Two points determine infinitely many circles (centres on the perp bisector). You need the third point to pin the centre.
FBISE Exam Check
- Two points determine infinitely many circles (centres on the perp bisector). You need the third point to pin the centre.
60-second revision
- Three non-collinear points lie on exactly one circle (their circumcircle).
- UNIQUE CIRCUMCIRCLE: A, B, C not on a line: one circle.
- COLLINEAR FAIL: Three on a line have no finite circumcentre (perp bisectors are parallel or the same).
- Worked-method reminder: so O is the unique circumcentre
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.