Angle Bisector Theorem Calculator

Math Geometry • Basic Shapes and Properties

Written by STEM Calculators Team Published January 21, 2026 Updated February 24, 2026

Use the Angle Bisector Theorem to solve unknown segments on the opposite side of a triangle, or verify a given configuration.

Inputs accept 1e-3, pi, e, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Operation: Side \(AB\) (units): Side \(AC\) (units): Segment \(BD\) (units): Segment \(DC\) (units):

Optional: If you know total \(BC\) (for “Split \(BC\)” operation)

Side \(BC\) (units):

For split mode: \(BD=\dfrac{AB}{AB+AC}\,BC\), \(DC=\dfrac{AC}{AB+AC}\,BC\).

Advanced (university): bisector length \(AD\)

Compute \(AD\):

Uses \(AD^2 = AB\cdot AC\left(1-\dfrac{BC^2}{(AB+AC)^2}\right)\) (requires a feasible triangle).

Diagram controls

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Pan/zoom: drag to pan • wheel/trackpad to zoom • pinch on touch • “Reset view” restores fit.

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Enter values and click Calculate.

Triangle & angle bisector diagram (pan/zoom enabled)

The point \(D\) is placed using the computed (or provided) segments \(BD\) and \(DC\).

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8. Angle Bisector Theorem Calculator

In a triangle \(ABC\), let \(AD\) be the internal angle bisector of \(\angle A\), meeting the opposite side \(BC\) at \(D\). The Angle Bisector Theorem relates how the bisector divides the opposite side to the lengths of the adjacent sides.

The theorem

If \(AD\) bisects \(\angle A\), then:

\[ \begin{aligned} \frac{BD}{DC} &= \frac{AB}{AC} \end{aligned} \]

How to solve common unknowns

Depending on what is unknown, rearrange the ratio:

\[ \begin{aligned} BD &= DC\cdot \frac{AB}{AC} \\ DC &= BD\cdot \frac{AC}{AB} \\ AB &= AC\cdot \frac{BD}{DC} \\ AC &= AB\cdot \frac{DC}{BD} \end{aligned} \]

Alternative form: split the whole side \(BC\)

If you know the full opposite side \(BC\) instead of one segment, you can compute both segments directly:

\[ \begin{aligned} BD &= \frac{AB}{AB+AC}\,BC \\ DC &= \frac{AC}{AB+AC}\,BC \end{aligned} \]

Worked example (triangle-feasible)

Given adjacent sides \(AB=7\) cm, \(AC=10\) cm, and one segment \(DC=9\) cm, find the other segment \(BD=x\).

\[ \begin{aligned} \frac{BD}{DC} &= \frac{AB}{AC} \\ BD &= DC\cdot \frac{AB}{AC} \\ &= 9\cdot \frac{7}{10} \\ &= 6.3\ \text{cm} \end{aligned} \]

Consistency check

If all four values are given, the configuration is consistent with an angle bisector when the two ratios match:

\[ \begin{aligned} \frac{BD}{DC} \stackrel{?}{=} \frac{AB}{AC} \end{aligned} \]

Advanced (university): bisector length

If the three side lengths form a feasible triangle, the bisector length \(AD\) can be computed using:

\[ \begin{aligned} AD^2 &= AB\cdot AC\left(1-\frac{BC^2}{(AB+AC)^2}\right) \end{aligned} \]

Common mistakes

Mixing up which sides are adjacent to the bisected angle (use \(AB\) and \(AC\) for \(\angle A\)).
Using \(BD/DC = AC/AB\) (the ratio is \(AB/AC\) when bisecting \(\angle A\)).
Entering negative or zero lengths (segments and sides must be positive).

Tip: the diagram in the calculator places \(D\) using the segments \(BD\) and \(DC\). If the side lengths cannot form a real triangle, the tool will still compute segment ratios, but the diagram may be shown as “not to scale”.

Frequently Asked Questions

What is the angle bisector theorem in a triangle?

If AD bisects angle A in triangle ABC and meets BC at D, then the opposite side is divided proportionally: BD/DC = AB/AC. This links the two segments on BC to the two sides adjacent to angle A.

How do you solve for BD or DC using the angle bisector theorem?

Start from BD/DC = AB/AC and rearrange for the unknown. For example, BD = DC x (AB/AC) and DC = BD x (AC/AB).

How do you split BC into BD and DC if BC is known?

Use the split form: BD = (AB/(AB+AC)) x BC and DC = (AC/(AB+AC)) x BC. These two segments add up to BC by construction.

How can I check if my numbers are consistent with an angle bisector?

Compute both ratios BD/DC and AB/AC and compare them. The configuration matches an internal angle bisector when the two ratios are equal (within rounding or tolerance).

When can the bisector length AD be computed?

AD can be computed when the triangle is feasible and the side lengths are defined, using AD^2 = AB x AC x (1 - BC^2/(AB+AC)^2). The calculator can derive BC from BD + DC in modes where BD and DC are available.

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Frequently Asked Questions

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