Bisection

Bisection is a way to divide something into two equal parts. In geometry, these two parts are called congruent. This means they have the same size and the same shape.

You might use a bisector to split a line or an angle. A bisector is simply the line that does the splitting. Most often, we look at segment bisectors or angle bisectors. There are also ways to bisect things in three-dimensional space. In that case, we use a flat surface called a bisector plane.

There are different ways to work with these lines. A segment bisector passes through the exact middle of a line. We call this middle point the midpoint. A perpendicular bisector is even more special. It meets the segment at a right angle. Every point on a perpendicular bisector is the same distance from both ends of the segment.

You can also bisect an angle. An angle bisector splits an angle into two equal parts. Every point on an angle bisector is the same distance from the sides of the angle. An angle only has one bisector.

In classical geometry, people used simple tools to find these lines. They used a straightedge and a compass. You can bisect a segment by drawing circles with a compass. You place the compass on the endpoints of the segment. You draw arcs of equal size that cross each other. The line through those crossing points is the perpendicular bisector.

To bisect an angle, you draw a circle centered at the vertex. This circle hits the two sides of the angle. Then, you draw two more circles from those new points. The place where these circles meet helps you draw the bisector.

Many interesting things happen when these lines meet. In a triangle, the three angle bisectors all meet at one single point. This special point is called the incenter.

There is also a rule called the angle bisector theorem. It relates the lengths of the sides of a triangle to the parts created by the bisector. In a rhombus, each diagonal bisects the opposite angles. Pierre Wantzel proved a famous fact about this. He showed that you cannot divide an angle into three equal parts using only a compass and ruler.

Bisection helps us find the center of many shapes. For example, the three perpendicular bisectors of a triangle meet at the circumcenter. This is the center of a circle that touches all three corners. In a triangle, the medians also bisect the sides. The medians meet at a point called the centroid. This point is the center of mass for the triangle.

Even complex shapes like parabolas use bisection. A tangent to a parabola can bisect an angle. Math uses these equal parts to find balance and symmetry everywhere.