Proof that the bisectors of the sides of a triangle meet

The perpendicular bisector of the side of a triangle is a line through the middle point of the side at right angles to (perpendicular to) the side itself. In this figure of a triangle with vertices A, B, and C, the perpendicular bisectors are marked with dotted lines.

We claim that, for any triangle, the three perpendicular bisectors of the sides meet at a point called the circumcentre.

Proof

The perpendicular bisector of a side such as AB consists of points which are equidistant (the same distance) from A and B.

Similarly the line bisecting AC is the set of points equidistant from A and C.

Thus the point where these two lines meet, O, is equidistant from A, B, and C.

Since it is equidistant from B and C, it lies on the perpendicular bisector of BC, and we have proved that the three perpendicular bisectors always meet each other.

The circumcircle

Since these are all the same distance from O, a circle of radius OA meets all three vertices of the triangle.