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Circles in a triangle

For any triangle we can draw a circle that is either circumscribed or inscribed. The circumcircle passes through all the vertices of the triangle, and the incircle touches all three sides of the triangle.

Circumcircle of a triangle

The circumcircle is the circle that passes through all the vertices of the triangle. This circle always exists, and there is only one. So we can say that through any three points that do not lie on one straight line there passes exactly one circle.

To draw the circumcircle of a triangle, we need the idea of a perpendicular bisector. The perpendicular bisector of side c is the line that is perpendicular to side c and passes through the midpoint of side c, the point Mc. How to find the midpoint of a side is described in the construction of medians. Once we have the midpoint of a side, we can draw a perpendicular through it and so create the perpendicular bisector.

Every triangle has three sides, so it also has three perpendicular bisectors. These meet at a single point, which we usually call O. This point is the centre of the circumcircle (the circumcentre). So we proceed as follows. We start with a triangle and want to draw its circumcircle:

An ordinary triangle

First we draw the perpendicular bisectors of all three sides. That means we find the midpoints of all three sides and then draw a perpendicular through each midpoint.

The perpendicular bisectors marked, meeting at one point

All that remains is to draw the circle. The circumcircle passes through all the vertices, so its radius is easy to find: it is the distance from the centre O to any vertex.

Circumcircle of a triangle

Incircle

A circle is inscribed in a triangle if it touches all three sides, that is, it has exactly one point in common with each side. To find the centre of the incircle we need the idea of an angle bisector, which is explained in a separate article. The centre of the incircle (the incentre) is the point where all the angle bisectors meet. So we start with an ordinary triangle and draw the angle bisectors:

Triangle with the angle bisectors

The point of intersection I is the centre of the incircle. We still need to find the radius. We draw a perpendicular to any side that passes through the point I.

Perpendicular to side c

The length of the segment IPc is equal to the radius of the incircle. Now all that remains is to draw the circle:

Circle k inscribed in triangle ABC

Chapters: Triangle, Altitude of a triangle, Median of a triangle, Circles in a triangle, Right-angled triangle, Constructing a triangle, Congruent triangles, Similar triangles, Area of a triangle, Pythagoras’ theorem