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Home Page > Math > Geometry Calculators

Angle Bisector Calculator

Enter three sides or three vertex coordinates in MiniWebtool’s Angle Bisector Calculator to calculate a triangle’s angle bisectors. Find bisector lengths, division points on opposite sides, incenter and inradius, with an interactive diagram and step-by-step formulas.

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Angle Bisector CalculatorTry it now — free ▼
The bisector splits the opposite side in the ratio of the other two

Enter three side lengths or three vertex coordinates below to get the three bisector lengths and where each meets the opposite side, with the steps.

Examples:
LIVE PREVIEW
Opposite vertex A
Opposite vertex B
Opposite vertex C
Examples:
LIVE PREVIEW
Vertex A
Vertex B
Vertex C

Embed Angle Bisector Calculator Widget

About Angle Bisector Calculator

The Angle Bisector Calculator computes the angle bisectors of any triangle. Enter three side lengths or three vertex coordinates, and the calculator finds all three bisector lengths, the points where each bisector meets the opposite side, the incenter, the inradius, and displays an interactive diagram. All computations include step-by-step MathJax formulas.

∠
Angle Bisector
A segment from a vertex to the opposite side that divides the angle into two equal halves.
◉
Incenter
The point where all three angle bisectors meet. It is the center of the inscribed circle.
⊙
Inradius
The radius of the inscribed circle, equal to the area divided by the semi-perimeter: r = K/s.
⚖
Bisector Theorem
The bisector from A divides side BC in the ratio AB:AC = c:b (proportional to adjacent sides).

At a Glance

The Angle Bisector Calculator finds the three angle bisectors of a triangle from its side lengths or vertex coordinates. It also gives division points, the incenter and the inradius, with an interactive diagram.

Angle Bisector Calculator: features at a glance
FeatureWhat you getGood for
Two input modesEnter three side lengths a, b and c, or the x and y coordinates of three vertices.Triangles given either by sides or by coordinates
Bisector lengthsGives all three bisector lengths with t_a = 2bc cos(A/2) / (b + c), or the equivalent formula that uses side lengths only.Geometry homework and construction problems
Division pointsShows where each bisector meets the opposite side, using the angle bisector theorem BD/DC = c/b.Locating the foot of each bisector
Incenter and inradiusGives the incenter as the side-weighted average of the vertices and the inradius r = area / semi-perimeter.Finding the center and size of the inscribed circle
Interactive diagramToggle bisectors, division points, the incircle, angle arcs and labels.Focusing on one property at a time
Preset trianglesQuick buttons load equilateral, right 3-4-5, isosceles 5-5-8, obtuse 3-4-6 and scalene examples.Trying the tool on a known triangle

Angle Bisector Formulas

Angle Bisector Calculator: Angle Bisector Formulas
PropertyFormulaDescription
Bisector Length (from A)\( t_a = \frac{2bc \cos(A/2)}{b+c} \)Length of the angle bisector from vertex A to side BC
Alternative Formula\( t_a = \frac{\sqrt{bc[(b+c)^2 - a^2]}}{b+c} \)Uses side lengths only, no trigonometry needed
Bisector Theorem\( \frac{BD}{DC} = \frac{c}{b} = \frac{AB}{AC} \)Division ratio of opposite side by the bisector
Division Segment\( BD = \frac{ac}{b+c} \)Length from B to division point D on BC
Incenter\( I = \frac{a \cdot A + b \cdot B + c \cdot C}{a+b+c} \)Weighted average of vertices using opposite side lengths
Inradius\( r = \frac{K}{s} \)Area K divided by semi-perimeter s

How to Use This Calculator

  1. Choose input mode: Select "Three Sides" if you know a, b, c, or "Three Vertices" if you have coordinates.
  2. Enter values: Type the three side lengths or the (x, y) coordinates for each vertex. Use the quick example buttons to try preset triangles.
  3. Click Calculate: Press the "Calculate Angle Bisectors" button to see results.
  4. Explore the diagram: Toggle layers (bisectors, division points, incircle, angle arcs, labels) to focus on specific properties.
  5. Review formulas: Scroll down to the step-by-step solution to see every formula with substituted values.

Understanding the Angle Bisector Theorem

The Angle Bisector Theorem is one of the fundamental results in triangle geometry. It states that if a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides. Specifically, if the bisector from vertex A meets side BC at point D, then BD/DC = AB/AC = c/b.

This theorem has many practical applications: it is used in triangle construction, in proving properties of the incircle, and in coordinate geometry problems. The angle bisector length formula \( t_a = \frac{2bc \cos(A/2)}{b+c} \) can be derived by applying the cosine rule to the two sub-triangles created by the bisector.

Properties of Angle Bisectors

Frequently Asked Questions

What is an angle bisector of a triangle?
An angle bisector is a line segment from a vertex to the opposite side that divides the vertex angle into two equal halves. Every triangle has three angle bisectors, and they all meet at a single point called the incenter, which is the center of the inscribed circle (incircle).
How do you calculate the length of an angle bisector?
The length of the angle bisector from vertex A to side BC is given by the formula: ta = (2bc × cos(A/2)) / (b + c), where b and c are the sides adjacent to angle A. An alternative formula that uses only side lengths is: ta = √(bc[(b+c)² − a²]) / (b + c).
What is the angle bisector theorem?
The angle bisector theorem states that an angle bisector of a triangle divides the opposite side into two segments proportional to the adjacent sides. For the bisector from A to side BC at point D: BD/DC = AB/AC = c/b. This is a fundamental theorem used in many geometric proofs and constructions.
Where do the three angle bisectors of a triangle meet?
The three angle bisectors of a triangle always meet at a single point called the incenter, denoted I. The incenter is the center of the inscribed circle (incircle) and is always located inside the triangle, regardless of whether the triangle is acute, right, or obtuse.
What is the difference between angle bisector and perpendicular bisector?
An angle bisector divides an angle into two equal parts and goes from a vertex to the opposite side. A perpendicular bisector divides a side into two equal parts at a 90° angle. The three angle bisectors meet at the incenter (center of inscribed circle), while the three perpendicular bisectors meet at the circumcenter (center of circumscribed circle).

Reference this content, page, or tool as:

"Angle Bisector Calculator" at https://MiniWebtool.com/angle-bisector-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-03

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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