Angle Bisectors in a Triangle
In any triangle, an angle bisector is a line (or segment) that starts at a vertex and splits that vertex angle into
two equal angles. If the bisector comes from vertex \(A\), it meets the opposite side \(\overline{BC}\) at a point \(D\). This simple
construction is surprisingly powerful because it connects angles, side lengths, and ratios in a precise way.
Angle Bisector Theorem
The core fact is the Angle Bisector Theorem:
\[
\frac{BD}{DC}=\frac{AB}{AC}.
\]
Using the standard triangle convention \(c=AB\) and \(b=AC\), this becomes \(\dfrac{BD}{DC}=\dfrac{c}{b}\). In other words, the angle
bisector from \(A\) divides the opposite side in the same ratio as the two sides adjacent to \(A\). This is why angle bisectors are
commonly used in constructions and proofs: they turn geometry into a clean proportional relationship.
Since \(BD+DC=BC=a\), you can solve explicitly for the two segments:
\[
BD=\frac{ac}{b+c},\qquad DC=\frac{ab}{b+c}.
\]
These expressions are especially useful when you need the exact location of the bisector intersection point on the base.
Bisector Length Formula
Beyond where the bisector hits the base, we often need the length of the bisector itself. The internal bisector from
vertex \(A\) is typically denoted \(l_a\). A convenient formula uses the half-angle:
\[
l_a=\frac{2bc\cos\!\left(\frac{A}{2}\right)}{b+c}.
\]
This version is ideal if you know the adjacent sides \(b\) and \(c\) and the angle \(A\). If you only know the three sides \(a\), \(b\),
and \(c\), you can still compute \(A\) from the Law of Cosines, then apply the formula above.
There is also a purely side-based equivalent form:
\[
l_a=\frac{\sqrt{bc\left((b+c)^2-a^2\right)}}{b+c}.
\]
Both formulas produce the same result; they are just different ways to package the same geometry.
Why It Matters
Angle bisectors are tightly connected to the incenter of a triangle—the unique point where all three internal angle
bisectors intersect. The incenter is the center of the inscribed circle (incircle), which touches all three sides. Because of that,
bisectors appear in problems involving tangency, equal distances to sides, triangle optimization, and classical Euclidean constructions.
This calculator focuses on the bisector from \(A\): it computes \(l_a\), computes the base split \(BD\) and \(DC\), and visualizes the
triangle with the bisector segment. The interactive diagram helps you see that as the triangle shape changes, the bisector still divides
\(\overline{BC}\) in the consistent ratio \(c:b\), and the computed length always matches the distance from \(A\) to the intersection
point \(D\).
