Construct a triangle's circumcircle from side bisectors
Locate the circumcenter by perpendicular bisectors · Draw a circle through all three vertices · Recognise on-boundary, outside and degenerate cases
Section E4.5 · Extended · Compass-and-straightedge constructions and scale drawings
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Perpendicular-bisector construction
- Equal-distance locus
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Locate the circumcenter by perpendicular bisectors
- Draw a circle through all three vertices
- Recognise on-boundary, outside and degenerate cases
Prepare your paper
When a practice task gives numerical input geometry rather than a printed figure, first prepare that source segment, angle, triangle or circle with measuring tools as needed. Then put the measuring scale or protractor aside for the requested formal copying or bisector steps: use a compass and unmarked straightedge to locate the new points. The preparation supplies the givens; it is not a replacement for the formal construction.
The website diagrams explain relationships; their screen size is not a centimetre calibration or a ready-to-measure template. Keep the input figure and useful construction arcs on paper. Your drawing is self-checked and needs human review for reliable marking.
Equal distances to vertices
Construct perpendicular bisectors of two sides of a non-degenerate triangle. Their intersection O is equidistant from all three vertices. Set the compass from O to any vertex and draw the circumcircle; it passes through every vertex.
The center is inside an acute triangle, at the hypotenuse midpoint of a right triangle, and outside an obtuse triangle. An outside center does not invalidate a circumcircle. It is different from the incircle's always-internal incenter.
Check the defining property
For the illustrated right triangle with vertices (0,0), (6,0), (0,8), the hypotenuse midpoint is (3,4) and its distance to every vertex is 5. The circumcircle therefore has radius 5.
Three distinct collinear points do not form a non-degenerate triangle or lie on one finite circle: a line meets a circle in at most two distinct points. Do not force a triangle-circle construction for that degenerate input.
Sources for this lesson
Pause and explain
Can an obtuse triangle have its circumcenter outside?
Worked example
Construct a circumcircle for the illustrated 6–8–10 right triangle.
Show the worked solution
- Construct two side perpendicular bisectors; they meet at (3,4).
- This is the hypotenuse midpoint, distance 5 from each vertex.
- Draw a radius-5 circle from that center.
Answer Circumcenter (3,4), radius 5, passing through all three vertices.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Describe a compass-and-straightedge circumcircle construction for a supplied triangle.
Give me a hint
Use the side equal-distance loci.
Compare my reasoning
- Construct perpendicular bisectors of two sides and locate their intersection O.
- Use OA as the radius; equal-distance properties give OA = OB = OC.
Circle centered at the side-bisector intersection, with a vertex distance as radius.
Look for these in your work
- Retain bisector arcs
- Check all three vertices
A right triangle has vertices (0,0), (4,0), (0,3). Find its circumcenter and radius.
Give me a hint
Use the midpoint of its length-5 hypotenuse.
Compare my reasoning
- Midpoint is (2,1.5).
- Half the hypotenuse gives radius 2.5; squared distance to (0,0) is 4+2.25 = 6.25.
Circumcenter (2,1.5); radius 2.5.
Look for these in your work
- Use the right-triangle property
- Verify a vertex distance
Three distinct points A, B and C lie on the same straight line. Can a finite circle pass through all three? Explain the construction failure.
Give me a hint
Check the non-degenerate triangle condition.
Compare my reasoning
- A line has at most two distinct intersections with a circle.
- The perpendicular bisectors of the collinear segments cannot locate a common finite center equidistant from all three.
No finite circumcircle; the supplied points are degenerate for this triangle construction.
Look for these in your work
- Check the input condition
- Do not invent a center from an approximate drawing
Common mistakes
- A circumcircle goes through vertices, not tangent to sides.
- Its center need not be inside the triangle.
What can you explain now?
Can an obtuse triangle have its circumcenter outside?
Compare with the explanation
Yes; the circumcircle still passes through all three vertices.
Equal vertex distances, not internal location, define this circle.
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Make it stick
Can an obtuse triangle have its circumcenter outside?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.