Angle Bisector Calculator

Calculate the angle bisectors of a triangle. Enter three sides or three vertex coordinates to find bisector lengths, division points on opposite sides, incenter, inradius, and see an interactive diagram with step-by-step formulas.

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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.

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Angle Bisector

A segment from a vertex to the opposite side that divides the angle into two equal halves.

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Incenter

The point where all three angle bisectors meet. It is the center of the inscribed circle.

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Inradius

The radius of the inscribed circle, equal to the area divided by the semi-perimeter: r = K/s.

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Bisector Theorem

The bisector from A divides side BC in the ratio AB:AC = c:b (proportional to adjacent sides).

Angle Bisector Formulas

Property	Formula	Description
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

Choose input mode: Select "Three Sides" if you know a, b, c, or "Three Vertices" if you have coordinates.
Enter values: Type the three side lengths or the (x, y) coordinates for each vertex. Use the quick example buttons to try preset triangles.
Click Calculate: Press the "Calculate Angle Bisectors" button to see results.
Explore the diagram: Toggle layers (bisectors, division points, incircle, angle arcs, labels) to focus on specific properties.
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

Every triangle has exactly three interior angle bisectors.
All three angle bisectors always intersect at a single point called the incenter.
The incenter is always located inside the triangle, regardless of the triangle type.
The incenter is equidistant from all three sides, and that distance is the inradius.
In an equilateral triangle, each angle bisector also serves as a median, altitude, and perpendicular bisector.
The longest angle bisector always comes from the vertex with the smallest angle.
The bisector length is always less than or equal to the geometric mean of the two adjacent sides.

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: t_a = (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: t_a = √(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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Angle Bisector Calculator

About Angle Bisector Calculator

Angle Bisector Formulas

How to Use This Calculator

Understanding the Angle Bisector Theorem

Properties of Angle Bisectors

Frequently Asked Questions

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