Construct a triangle's incircle from internal angle bisectors
Locate the incenter by internal angle bisectors · Use a perpendicular side distance as the radius · Distinguish tangency to sides from passing through vertices
Section Paper 4 · Historical Extended · calculator
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
- Angle-bisector construction
- Perpendicular through a point
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 incenter by internal angle bisectors
- Use a perpendicular side distance as the radius
- Distinguish tangency to sides from passing through vertices
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 sides
For a non-degenerate triangle, construct two internal angle bisectors. Their intersection I is the incenter, inside the triangle. Internal-bisector points are equidistant from the two arm lines, so I is equidistant from all three side lines.
Construct a perpendicular from I to one side, meeting it at F. Use IF as the compass radius to draw the incircle centered at I. It touches each triangle side; it does not generally pass through any vertex. Do not use the vertex distance as its radius.
A numerical check
The illustrated right triangle has perpendicular sides 6 and 8, hypotenuse 10, incenter (2,2) and radius 2. Its area is 24 and semiperimeter is 12. Dividing the area into three triangles with common height r shows area = r×semiperimeter, confirming r = 24/12 = 2.
This radius check is optional verification of the construction; locating the incenter still uses formal internal bisectors. External-bisector intersections can give other tangent circles outside the triangle, not its incircle.
Sources for this lesson
Pause and explain
What radius is used for a triangle's incircle?
Worked example
Construct the incircle of the illustrated 6–8–10 right triangle.
Show the worked solution
- Construct two internal angle bisectors to locate I.
- Construct the perpendicular IF to a side; its length is 2 in this triangle.
- Draw the circle centered at I with that radius; all three sides are tangent.
Answer Incenter (2,2), radius 2, with tangency to all three sides.
Try a topic test
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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 the formal incircle construction for a supplied non-degenerate triangle.
Give me a hint
Locate I before setting the radius.
Compare my reasoning
- Construct two internal angle bisectors and mark their intersection I.
- Construct a perpendicular from I to a side, then draw the I-centered circle using that distance.
Internal bisectors locate I; a perpendicular side distance supplies the radius.
Look for these in your work
- Leave bisector evidence
- Use the correct radius
A 3–4–5 right triangle has area 6 and semiperimeter 6. Check its incircle radius using area = r×semiperimeter.
Give me a hint
Divide area by semiperimeter.
Compare my reasoning
- r = 6/6 = 1.
- Its three area pieces have total r×6 = 6, confirming the given area.
Radius 1, consistent with the area check.
Look for these in your work
- Use the side-distance radius
- Separate checking from locating the center
A student places a circle center outside the triangle and makes the circle tangent to extensions of its sides. Does that prove it is the incircle?
Give me a hint
Check internal location and tangency to the actual sides.
Compare my reasoning
- No: an outside tangent circle is not the triangle's inscribed incircle.
- Use the internal angle-bisector intersection inside the triangle and its perpendicular side radius.
No. The incircle uses the internal incenter and touches the actual triangle sides.
Look for these in your work
- Distinguish sides from their extensions
- Check the center's location
Common mistakes
- Use perpendicular distance to a side, not distance to a vertex.
- Use internal bisectors for the incircle.
What can you explain now?
What radius is used for a triangle's incircle?
Compare with the explanation
The perpendicular distance from the incenter to a side.
It is the same for all three sides and gives tangency.
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Make it stick
What radius is used for a triangle's incircle?
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.