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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).
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Unique Circle Through Three Points
What is it?
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).
In simple words
- 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
Important terms
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'.
What you should learn
- 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.
Examples
- Triangle ABC
- Perp bisectors meet
- Not infinitely many
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'.
Recommended for Class 10
Full school-level explanation
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).
Collinear or not:
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.
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.
Remember
- equidistant centre
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'.
Remember
- need a triangle
Examples
- Three on a line: no unique circle
- Two points: infinitely many
- One point: even more
Remember
- 3 non-collinear -> unique circle
- OA=OB=OC
- two points are not enough
- equidistant centre
- need a triangle
- 3 non-collinear -> unique circle
- 3 non-collinear -> unique circle
- mathematical reasoning
How the ideas connect
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
Deeper understanding
Go beyond memorising Unique Circle Through Three Points. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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 mistakes
Exam focus
Core Idea
UNIQUE CIRCUMCIRCLE
COLLINEAR FAIL
Worked Method
Exam focus
Quick revision
- 3 non-collinear -> unique circle
- OA=OB=OC
- two points are not enough
- equidistant centre
- UNIQUE CIRCLE THROUGH THREE POINTS: Three non-collinear points lie on exactly one circle (their circumcircle). Three collinear points do
- UNIQUE CIRCUMCIRCLE: A, B, C not on a line: one circle. Its centre is equidistant from A, B, and C (the circumradius). Th
Two points determine infinitely many circles (centres on the perp bisector). You need the third point to pin the centre.
For long answers on Unique Circle Through Three Points: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- 3 non-collinear -> unique circle
- OA=OB=OC
- two points are not enough
- equidistant centre
Important formulas
- 3 non-collinear -> unique circle
- OA=OB=OC
- two points are not enough
Common exam traps
- Two points determine infinitely many circles (centres on the perp bisector). You need the third point to pin the centre.
- Not infinitely many
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Two points determine infinitely many circles (centres on the perp bisector). You need the third point to pin the centre.
- Mistake
Not infinitely many
- Mistake
Do not treat UNIQUE CIRCUMCIRCLE and COLLINEAR FAIL as the same step; each has a different purpose.
- Mistake
Two points determine infinitely many circles (centres on the perp bisector). You need the third point to pin the centre.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
3 non-collinear -> unique circle
OA=OB=OC
two points are not enough
equidistant centre
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 pe
OA=OB=OC
- 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 pe
- Formula
- OA=OB=OC
- Example
- Triangle ABC
- Critical warning
- Two points determine infinitely many circles (centres on the perp bisector). You need the third point to pin the centre.
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Exam Practice
Define
Q1. Define Unique Circle Through Three Points.
Explain
Q2. Explain: UNIQUE CIRCUMCIRCLE.
Example
Q3. Give one clear example of Unique Circle Through Three Points and say why it fits.
Mistake
Q4. State one common mistake students make about Unique Circle Through Three Points.
Exam note
Q5. Write a short exam note on Unique Circle Through Three Points: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.