Triangle Angle Bisectors

A bisector of a triangle is a segment that splits one of its interior angles into two equal parts and extends to the opposite side.

Imagine you have a triangle. Pick one of its interior angles, say angle alpha, and divide it right down the middle—like cutting a slice of pizza perfectly in half.

The segment AD that you just drew is called a bisector.

Since every triangle has three angles, you can draw three interior bisectors.

Here’s the cool part: all three bisectors intersect at a single point! This special point is called the incenter.

So why does the incenter matter? It’s actually the center of the incircle—the largest circle that fits perfectly inside the triangle, touching all three sides without crossing them. In other words, the incircle is tangent to each side of the triangle.

The Bisector Proportionality Theorem

A bisector doesn’t just split an angle—it also has another interesting property. When it reaches the opposite side, it divides it into two segments that are proportional to the other two sides of the triangle.

In simple terms, if we call these two segments \(m\) and \(n\), and the two sides forming the angle \(a\) and \(b\), then the following relationship holds:

\[ \frac{m}{n} = \frac{a}{b} \]

Think of it as the bisector maintaining a natural balance between the sides.

Let’s take a look at a practical example.

In this triangle, the bisector of angle \( \beta \) divides the opposite side into two segments, \( m \) and \( n \).

The ratio between the segment lengths \( m \) and \( n \) (smaller segment divided by the larger one) is approximately 0.7454:

$$ \frac{m}{n} = \frac{3.4164}{4.5836} = 0.7454 $$

Interestingly, the ratio between the sides \( AB \) and \( BC \), which form angle \( \beta \), is also 0.7454, as long as we place the smaller side in the numerator:

$$ \frac{AB}{BC} = \frac{6.3246}{8.4853} = 0.7454 $$

This property holds for all triangle bisectors.

External Bisectors

Besides the internal bisectors, a triangle also has external bisectors. Instead of splitting an interior angle, these divide an exterior angle of the triangle into two equal parts.

If you draw two external bisectors from different angles and combine them with the internal bisector of the remaining angle, you’ll notice they meet at a point outside the triangle. This point is called the excenter.

So what’s special about the excenter? It’s the center of another unique circle—the excircle. This circle is tangent to one side of the triangle and to the extensions of the other two sides.

In a way, you can think of the excircle as the incircle’s “external cousin.”