Angle bisector theorem

In geometry, the angle bisector theorem helps us learn about lines in a triangle. This theorem talks about a special line that cuts an angle in a triangle into two equal parts.

When this line cuts the angle, it makes two smaller parts on the side across from that angle. The theorem tells us that the sizes of these two smaller parts are related to the sizes of the triangle’s other two sides.

In simple terms, the angle bisector theorem says that the ratio, or comparison, of the lengths of these two smaller parts is the same as the ratio of the lengths of the other two sides of the triangle. This rule is useful for solving many geometry problems, like finding unknown side lengths or proving facts about shapes.

This idea is important because it links angles and sides in triangles. It is a handy tool for math problems and real-life projects that involve triangles.

Theorem

Imagine a triangle ABC. If we draw a line from vertex A that splits the angle at A exactly in half, and this line meets the side BC at a point D, the angle bisector theorem tells us something interesting. It says that the ratio of the lengths BD to CD is the same as the ratio of the lengths AB to AC.

This theorem can also be used in reverse. If a point D on side BC splits it in the same ratio as the sides AB and AC, then the line AD is the angle bisector of angle A. This idea helps when we know the angles and side lengths and need to solve problems or prove something about the triangle.

Proofs

There are many ways to prove the angle bisector theorem. One common way uses similar triangles. When you reflect a triangle across a line that is perpendicular to the angle bisector, you make new triangles that look the same as each other. Because similar triangles have sides that match in size, this shows the sides of the first triangle are split in a special way.

Another way uses the law of sines. By using this rule for the smaller triangles made by the angle bisector, you can show the side lengths match what the theorem says. This works because some angles in these triangles are the same or add up to 180 degrees, which helps us compare their sides.

Other proofs use ideas like triangle heights or areas. For example, looking at the areas of the smaller triangles made by the angle bisector can also show the same pattern between the sides. Each method helps us see why the angle bisector theorem is true.

Main article: Angle bisector theorem

Length of the angle bisector

The length of an angle bisector in a triangle can be found with a special formula. This formula connects the lengths of the triangle's sides and the parts made when the angle bisector splits the opposite side.

The formula uses a value called k, which comes from the angle bisector theorem. By using Stewart's theorem, a helpful idea in geometry, we can find the exact length of the angle bisector. This shows how the sides and the parts are related in a triangle when an angle is split in half.

Exterior angle bisectors

When we look at the exterior angles of a triangle that isn’t equilateral, there are special rules about how the sides relate in length. If we draw a line that splits an exterior angle at one corner of the triangle and extend it to meet the opposite side, this creates a point. Doing this for each corner gives us three such points.

These three points — where the exterior angle bisectors meet the extended sides of the triangle — all lie on a single straight line. This shows an interesting property of how angles and sides in a triangle are connected.

History

The angle bisector theorem was known a long time ago. It is in Proposition 3 of Book VI of Euclid's Elements. After that, mathematicians studied how the theorem works with angle bisectors inside and outside shapes.

Applications

The angle bisector theorem is useful in geometry. It helps find the incenter of a triangle. The incenter is where the triangle's angle bisectors meet. The theorem is also used with the Circles of Apollonius. These are important curves in geometry.