Every circle construction starts by locating a centre
This chapter has no new theory. It converts the circle theorems you already know into drawing instructions, and the whole of it rests on one habit: before drawing any circle, find the point that must be its centre.
The centre is always defined by a distance condition, and a distance condition is a locus. Two loci fix a point, and once you have the point, one compass setting finishes the job.
| Circle wanted | Centre lies on | Because |
|---|---|---|
| Through 3 points | the perpendicular bisectors of the joining chords | equidistant from two points ⟹ on their perpendicular bisector |
| Touching 3 sides of a triangle | the angle bisectors | equidistant from two lines ⟹ on the angle bisector |
| Touching a line at a given point | the perpendicular to the line at that point | the tangent is perpendicular to the radius |
| Given radius through 2 points | arcs of that radius from each point | the centre is r from both |
Circumcircle and incircle
The circumcircle passes through all three vertices of a triangle. Its centre, the circumcentre, is equidistant from the three vertices, so it lies where the perpendicular bisectors of the sides meet.
The incircle touches all three sides. Its centre, the incentre, is equidistant from the three sides, so it lies where the angle bisectors meet. The radius is the perpendicular distance from that point to any side — you must drop that perpendicular to find it, not measure to a vertex.
- Circumcentre — perpendicular bisectors of the sides. Falls outside the triangle when the triangle is obtuse, and exactly on the hypotenuse when it is right-angled.
- Incentre — angle bisectors. Always inside the triangle, whatever its shape.
- For a right-angled triangle the circumcircle has the hypotenuse as its diameter — which is the angle-in-a-semicircle theorem read backwards.
The Equal from 2 points option is the circumcentre construction, and Equal from 2 lines is the incentre construction. Every circle construction in this chapter is one of these two, used twice.
Tangent constructions
Two tangent constructions are on the syllabus, and both come straight from the tangent–radius right angle.
To draw a tangent at a point T on the circle: join OT and construct the perpendicular to OT at T. That perpendicular is the tangent.
To draw the tangents from an external point P: join OP and find its midpoint M. Draw a circle centred M with radius MP. It cuts the original circle at two points, and each of those is a point of contact. The reason is the angle in a semicircle — any point on the circle with diameter OP sees OP at 90°, which is exactly the tangent–radius right angle you need.
Construct the tangents to a circle of radius 3 cm from a point P that is 7 cm from the centre O, and calculate their length.
- Draw the circle centre O, radius 3 cm, and mark P with OP = 7 cm.P is outside the circle since 7 > 3, so two tangents exist.
- Bisect OP to find its midpoint M, using equal arcs from O and from P.The perpendicular bisector construction gives M, and the arcs are the evidence for the mark.
- With centre M and radius MP = 3.5 cm, draw a circle. It cuts the first circle at A and B.Every point on this second circle sees OP at 90°, so ∠OAP = 90° and PA must be a tangent.
- Join PA and PB. These are the two tangents.
- Length: triangle OAP is right-angled at A, so
PA = √(7² − 3²) = √40 = 6.32cm.Measuring your drawing should give the same value to within a millimetre, which checks the construction.
Two tangents, each of length √40 ≈ 6.32 cm
The construction is the answer, not the picture
Marks in this chapter are for the arcs: the bisector arcs, the arcs locating the centre, the arcs of the second circle. A beautifully drawn tangent placed by eye scores nothing. Keep every arc, use a sharp pencil, and label the points you construct.
Checking a construction before you hand it in
Constructions are the one place in a maths paper where you can verify your own answer by measuring. Two quick checks catch nearly every error.
For a circumcircle, measure from your centre to all three vertices — the three distances must agree. For an incircle, drop a perpendicular to each side and check those three distances agree. If they do not, one of your bisectors is out and you can redraw it before losing the marks.
Before you leave this chapter
- Equidistant from points → perpendicular bisectors. Equidistant from lines → angle bisectors.
- Circumcentre = perpendicular bisectors of the sides; may fall outside the triangle.
- Incentre = angle bisectors; always inside. Its radius is the perpendicular distance to a side.
- Tangent at T: construct the perpendicular to OT at T. Tangents from P: draw the circle on OP as diameter.
- Leave all construction arcs visible — they are what the marks are for.
Circles that touch each other
Two circles that meet at exactly one point are said to touch, and the point of contact always lies on the straight line joining the two centres. That single fact answers every question in this part of the syllabus.
They can touch externally, sitting outside one another, in which case the distance between the centres is the sum of the radii. Or one can sit inside the other and touch internally, in which case the distance is the difference of the radii.
Using it to construct
To draw a circle of radius 3 cm touching a given circle of radius 5 cm externally, mark a point 8 cm from the given centre and use it as the new centre. For internal contact, mark a point 2 cm away instead. The construction is a single compass arc once you have decided which of the two cases the question describes.