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.
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
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"Angle Bisector Calculator" at https://MiniWebtool.com// from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-03
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