The circumcircle and the incircle

The circumcircle goes through the vertices, the incircle touches the sides. Perpendicular bisectors and angle bisectors find their centres.

One circle around, another inside

Every triangle has two special circles.

The circumcircle goes through all three vertices — the triangle sits in it like a picture in a round frame. Its centre is equally far from all three vertices.

The incircle is the biggest circle that fits inside — it touches all three sides. Its centre is equally far from all three sides.

How do you find those centres? Two lines you may know help. The perpendicular bisector of a side — a perpendicular through the midpoint of a side; every point on it is equally far from both ends of the side. The angle bisector — a ray that halves an angle; every point on it is equally far from both arms.

The centre of the circumcircle = the intersection of the perpendicular bisectors of the sides. The centre of the incircle = the intersection of the angle bisectors. Today you will construct both circles — and you will see that in an obtuse triangle the circumcentre moves outside.

Jonáš designs a logo for the school magazine: a triangle in a circle. He draws a circle by eye around the triangle — once a vertex sticks out, next time the circle is too big. Ema borrows a compass: “You must not guess the centre. It must be equally far from all three vertices.” She constructs the perpendicular bisectors of two sides, plants the intersection, opens the compass to one vertex and draws the circle. It goes through all three vertices first time. “Geometry is not drawing by eye,” she laughs, “it is looking for points that satisfy something.”

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Two bisectors are enough — the third will go through the same intersection, use it only as a check. Save time: two perpendicular bisectors of sides for the circumcircle, two angle bisectors for the incircle.

Constructing both circles

The circumcircle:

Step 1: construct the perpendicular bisector of side AB — with a compass equal arcs with centres A and B (radius bigger than half the side), above and below the side; the join of the intersections is the bisector. Step 2: construct the perpendicular bisector of side BC the same way. Step 3: mark the intersection of the bisectors S. It is equally far from A, B and C. Step 4: open the compass from S to any vertex and draw the circle. It must go through all the vertices.

Where the centre sits: in an acute triangle inside, in a right-angled one exactly at the midpoint of the hypotenuse, in an obtuse one outside. The circle always exists, only its centre travels.

The incircle:

Step 1: construct the bisector of the angle at vertex A — with a compass an arc across both arms, from the intersections two equal arcs inside the angle, the join with the vertex is the bisector. Step 2: the bisector of angle B the same way. Step 3: mark the intersection S — it is equally far from all the sides. Step 4: the radius = the perpendicular distance of S from any side (drop a perpendicular with a mark). Draw the circle; it should only touch the sides, not cut them.

Carpenters use the circumcircle (a round table around a triangular top), anyone who wants to cut the biggest circle into a triangle uses the incircle.

The circumcircle of an acute triangle

A triangle with sides 7, 6 and 5 cm.

I construct the perpendicular bisector of side AB: arcs with centres A and B (compass more than 3.5 cm), intersections above and below, I join. The same for BC. The intersection of the bisectors is S — it sits inside the triangle.

Compass from S to A, I draw. The circle goes through B and C too.

Check: measure |SA|, |SB|, |SC| — three equal distances. If one slips, the bisector was crooked; most often because of too small a compass opening when drawing the arcs.

A right-angled triangle: the circumcentre on the hypotenuse

Draw a right-angled triangle with legs 6 and 8 cm (right angle with a mark).

Construct the perpendicular bisectors of two sides. The intersection comes out exactly at the midpoint of the hypotenuse — with right-angled triangles it is always so.

The radius is half the hypotenuse: the hypotenuse measures 10 cm, the radius 5 cm.

This property can be read backwards too: when a vertex of a triangle sits on a circle and the opposite side is a diameter, the angle at that vertex is right. An old Greek trick that still works today.

The incircle: the biggest circle inside

A triangle with sides 8, 7 and 6 cm.

I construct the bisector of angle A: an arc cuts both arms, from the intersections equal arcs inward, I join with A. The same at vertex B. The intersection is S.

Radius: from S I drop a perpendicular (with a mark!) onto side AB, I measure the distance with a compass. I draw the circle.

Check: the circle only touches all three sides. If it cuts one, the radius was not perpendicular — the distance of a point from a line is always measured along a perpendicular, a slanted join is longer.

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Do not mix the bisectors: sides for the circumcircle, angles for the incircle. A hint: the circumcircle goes through the vertices — and the perpendicular bisector of a side watches the same distance from the vertices. The incircle touches the sides — and the angle bisector watches the same distance from the sides (the arms).

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.

Circles are drawn. Here calculate the angles, so you know the check.

Practise the questions

On paper: two circles for every triangle

Draw with a sharp pencil and have patience with the arcs — the accuracy of the bisectors decides everything.

Draw a triangle with sides 9, 7 and 6 cm and construct the circumcircle. Check with three distances from the centre.
For the same (or a new) triangle construct the incircle. Drop the radius as a perpendicular with a mark.
Draw an obtuse triangle and construct the circumcircle. Describe where the centre ended up.
Draw a right-angled triangle with legs 3 and 4 cm, construct the circumcircle and check that the centre sits at the midpoint of the hypotenuse and the radius is 2.5 cm.

Now you 💪

I know the difference: the circumcircle goes through the vertices, the incircle touches the sides.
I construct the circumcircle from perpendicular bisectors of the sides, the incircle from angle bisectors.
I know that the radius of the incircle is measured perpendicular to a side.

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Done when: In your book there are four constructions: circumcircle, incircle, circumcircle of an obtuse triangle (centre outside) and a right-angled one with the centre on the hypotenuse.

What to take from this lesson

Circumcircle: through the vertices, centre at the intersection of the perpendicular bisectors of the sides.
Incircle: touches the sides, centre at the intersection of the angle bisectors.
Two bisectors are enough, the third is a check.
In a right-angled triangle the circumcentre is the midpoint of the hypotenuse.
The distance of a point from a line is measured along a perpendicular.