Perpendicular Bisector of a Triangle

A perpendicular bisector of a side of a triangle is a line that:

• bisects the side (divides it into two equal parts), and
• is perpendicular to that side (meets it at a right angle).

It does not necessarily pass through any vertex of the triangle.

     A
    / \
   /   \
  B-----C
       |
       |  (perpendicular bisector of BC)
       |

Definition: The perpendicular bisector of a side of a triangle is the line that divides the side into two equal segments and forms a 90° angle with it.

If \( D \) is the midpoint of side \( BC \), and line \( l \) passes through \( D \) such that:

\( l \perp BC \)

then \( l \) is the perpendicular bisector of \( BC \).

• Every triangle has three perpendicular bisectors (one for each side).
• Each perpendicular bisector contains all the points that are equidistant from the endpoints of that side.
• A perpendicular bisector does not necessarily pass through any vertex.
• The perpendicular bisectors of a triangle always meet at one point (the circumcentre).

The three perpendicular bisectors of a triangle intersect at a special point called the circumcentre.

The circumcentre has important properties:

• It is equidistant from all three vertices of the triangle.
• A circle drawn with the circumcentre as centre and this common distance as radius passes through all three vertices.
• This circle is called the circumcircle of the triangle.

To construct the perpendicular bisector of a side \( BC \):

1. Find the midpoint \( D \) of \( BC \).
2. Use a compass to draw arcs above and below \( BC \) from both \( B \) and \( C \).
3. The two arcs intersect at two points—join them to form the perpendicular bisector.

Perpendicular bisectors help in:

• Finding the circumcentre of a triangle.
• Constructing circumcircles.
• Proving geometric relations and symmetry.
• Designing stable triangular structures.

The equidistant property makes perpendicular bisectors extremely important in geometry.