Angle bisector theorem

In geometry, the angle bisector theorem helps us understand how a line that cuts an angle in a triangle into two equal parts affects the lengths of the sides. When a line bisects, or divides, an angle, it creates two smaller segments on the side opposite that angle. The theorem tells us that the lengths of these two segments are related to the lengths of the other two sides of the triangle.

Specifically, the angle bisector theorem states that the ratio of the lengths of these two segments is the same as the ratio of the lengths of the other two sides of the triangle. This useful rule helps solve many problems in geometry, such as finding missing side lengths or proving geometric properties.

The concept is important because it connects angles and side lengths in triangles, providing a powerful tool for both theoretical and practical applications. Whether you're studying for a math test or working on a design project, the angle bisector theorem can help you understand the relationships within triangular shapes.

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 method uses similar triangles. When you reflect a triangle across a line that is perpendicular to the angle bisector, you can create new triangles that are similar to each other. Because similar triangles have sides in proportion, this shows the sides of the original triangle are divided in a specific ratio.

Animated illustration of the angle bisector theorem.

Another method uses the law of sines. By applying this rule to the triangles formed by the angle bisector, you can show that the ratios of the sides match what the theorem states. This works because certain angles in these triangles are equal or supplementary, which helps compare their sides.

Other proofs use ideas like triangle altitudes or areas. For example, looking at the areas of triangles formed by the angle bisector can also show the same relationship between the sides. Each method gives a clear picture of why the angle bisector theorem works.

Main article: Angle bisector theorem

| A B | | B D | = sin ⁡ ∠ A D B sin ⁡ ∠ D A B {\displaystyle {\frac {|AB|}{|BD|}}={\frac {\sin \angle ADB}{\sin \angle DAB}}}

| A C | | C D | = sin ⁡ ∠ A D C sin ⁡ ∠ D A C {\displaystyle {\frac {|AC|}{|CD|}}={\frac {\sin \angle ADC}{\sin \angle DAC}}}

Length of the angle bisector

The length of an angle bisector in a triangle can be calculated using a special formula. This formula connects the lengths of the sides of the triangle and the segments created when the angle bisector divides the opposite side.

The formula uses a constant called k, which comes from the angle bisector theorem. By using Stewart's theorem, which is a useful tool in geometry, we can find the exact length of the angle bisector. This shows how the sides and the segments relate to each other in a triangle when an angle is bisected.

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 bisects 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 first described in ancient times. It appears as Proposition 3 of Book VI in Euclid's Elements. Later, mathematicians built on this idea, exploring how the theorem works with both internal and external angle bisectors.

Applications

The angle bisector theorem has been useful in proving many important geometry results. For example, it helps find the coordinates of the incenter of a triangle, which is the point where the triangle's angle bisectors meet. It is also used in studying the Circles of Apollonius, which are important curves in geometry.