Calculate all three angle bisector lengths of a triangle from its sides. Find the incircle radius, incenter coordinates, and segment ratios using the angle bisector theorem. Includes presets and re...
An angle bisector of a triangle 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 — the center of the inscribed circle (incircle) that is tangent to all three sides.
The length of an internal angle bisector depends on the two adjacent sides and the side opposite the bisected angle, so bisector problems naturally connect side ratios, incenter geometry, and incircle behavior. This calculator is designed to show those relationships together instead of treating each bisector as an isolated value.
Angle bisectors are used in drafting, geometric construction, and proof work whenever you need to split an angle exactly or locate the incenter from side data alone. By reporting all three bisectors, the inradius, and the side-splitting ratios together, the calculator gives a fuller picture of how the triangle is structured.
This calculator is useful when you need more than a single bisector length. It shows all three internal angle bisectors together with the inradius, incenter coordinates, and the side-splitting ratios from the angle bisector theorem, so you can see how the full triangle geometry changes as the side lengths change. That makes it practical for geometry homework, proof checking, triangle drafting, and any design task where an inscribed circle or balanced interior point matters.
Angle bisector from A: t_a = √[bc((b+c)² − a²)] / (b+c) Angle bisector from B: t_b = √[ac((a+c)² − b²)] / (a+c) Angle bisector from C: t_c = √[ab((a+b)² − c²)] / (a+b) Angle bisector theorem: BD/DC = AB/AC = c/b Inradius: r = Area / s, where s = (a+b+c)/2 Incenter: I = (a·A + b·B + c·C) / (a+b+c)
Result: t_a ≈ 4.22, t_b ≈ 3.33, t_c ≈ 2.24, inradius = 1.00
With sides 3, 4, and 5, the semi-perimeter is 6 and Heron's formula gives area 6, so the inradius is 6 / 6 = 1. The bisector from vertex A has length √[4·5·((4+5)^2−3^2)] / 9 = √1440 / 9 ≈ 4.22, while the other two bisectors come out shorter because they start from smaller angles.
A triangle usually has three different angle bisector lengths because each one depends on the two adjacent sides and the angle at its starting vertex. In a scalene triangle, the bisector from the largest angle is often the longest route toward the opposite side, while an isosceles triangle forces two bisectors to match by symmetry. Looking at all three together is more informative than computing one in isolation because it shows how balanced or stretched the triangle really is.
All three internal angle bisectors meet at the incenter, the point that is the same distance from every side. That makes the incenter the natural anchor for the incircle, so bisector calculations are closely tied to finding the largest circle that fits inside the triangle. When the calculator reports the inradius and incenter coordinates alongside the bisector lengths, you can move directly from a side-length problem to an inscribed-circle interpretation without extra construction work.
The angle bisector theorem turns each bisector into a ratio tool: the point where it lands on the opposite side splits that side in proportion to the two adjacent sides. This is especially useful in proof problems, coordinate geometry, and drafting tasks where you need to divide a side accurately without measuring the interior angle itself. By comparing the computed segment lengths and the side ratio side by side, you can verify theorem-based work quickly and catch mistakes in hand calculations.
The angle bisector theorem states that the bisector from a vertex divides the opposite side into two segments whose ratio equals the ratio of the two adjacent sides: BD/DC = AB/AC. Use this as a practical reminder before finalizing the result.
Use the formula t_a = √[bc((b+c)² − a²)] / (b+c), where a is the opposite side and b, c are the adjacent sides. Keep this note short and outcome-focused for reuse.
The incenter is the point where all three angle bisectors meet. It is equidistant from all three sides and is the center of the inscribed circle (incircle).
Yes. Unlike the circumcenter or orthocenter, the incenter is always located inside the triangle for all triangle types.
The inradius r = Area / s, where s is the semi-perimeter. This gives Area = r · s, a useful identity for computing triangle area from the incircle.
Only in an equilateral triangle do the angle bisectors, medians, altitudes, and perpendicular bisectors all coincide. Apply this check where your workflow is most sensitive.