Excircles and excentres

The incircle of a triangle is fairly well known as the circle inscribed in the triangle – it’s tangent to each of the three sides, which means that it touches each one but doesn’t cross any of them. But did you know that triangles have three more circles that are (sort of) tangent to all three sides?

These circles are called the excircles, and whereas the incircle always lies completely inside the triangle, the excircles lie completely outside. ‘Wait a minute,’ I hear you protest. ‘How can they be tangent to all three sides if they’re outside the triangle?’

Well, we cheat a little bit. Each excircle is tangent to one side, and also tangent to extensions of the other two sides.

Extending a side of a triangle means making it longer by continuing it in the same direction. The extensions of the sides of triangle $ABC$ are shown as dashed lines in the image below.

Image of a triangle, ABC, with each of the sides extended by dashed lines

With the sides extended, there are now two pairs of angles at each corner. To draw the excircles, we need to find the angle bisectors (the lines that cut the angles in half) of each of these pairs. At each corner there will be an internal angle bisector, which cuts the angle inside the triangle in half, and an external angle bisector, which cuts the angles outside but touching the triangle in half.

Image of triangle ABC, with the extended sides shown as dashed lines. The angle inside A is cut in half with a line labelled 'Internal angle bisector. The other angle at A is cut in half with a line labelled 'External angle bisector'

The internal angle bisector of one corner and the external angle bisectors of the other two corners always cross at a single point, called an excentre. Every triangle has three excentres, and each one is the centre of an excircle.

Image of triangle ABC showing the internal and external angle bisectors of A, B and C. The three points where these lines cross are labelled 'Excentre'

Once we’ve found the excentres, there’s one more step to find the excircles. For each side of the triangle, we need to find a line perpendicular to it that goes through the excentre next to it. The excircle touches the triangle at the point where this line crosses the side of the triangle.

Image of triangle ABC showing the Excentres, lines through the excentres that are perpendicular to the sides of the triangle, and the Excircles.

Do you want to know something amazing about how excircles are related to the nine-point circle, which I wrote about a few weeks ago?

The nine-point circle is tangent to all three excircles!

Use the Geogebra applet below to explore how the excircles and nine point circle change when you change the shape of the triangle.

Some questions to think about:

  • When are two of the excircles the same size? When are they all the same size?
  • Can the nine point circle ever be bigger than an excircle? Can it ever be bigger than all three excircles?





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Hello! I’m Sam Hartburn, a freelance maths author, editor and animator. I also dabble in music and write mathematical songs. Get in touch by emailing or using any of the social media buttons above.

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