Why are the vertices weighted by the opposite sides?

The angle bisectors cut the opposite side in the ratio of the adjacent sides, and the weighted average is the coordinate form of that theorem. Averaging the vertices equally yields the centroid, a different point.

How does the inradius follow from the coordinates?

With r = area ÷ semi-perimeter. For the 3-4-5 triangle the area is 6 and the semi-perimeter 6, so r = 1, matching the distance from (1, 1) to either leg.

Does the incenter coincide with another center in an equilateral triangle?

Yes. In an equilateral triangle the incenter, centroid, circumcenter and orthocenter are the same point, one third of the height above the base.

📍 Incenter Calculator

Calculate the incenter (center of inscribed circle) coordinates of a triangle.

Vertex A:x₁ y₁

Vertex B:x₂ y₂

Vertex C:x₃ y₃

What is the Incenter?

The point where the three angle bisectors of a triangle meet. The incenter is equidistant from all three sides, and this distance is the inradius (r).

Incenter Coordinate Formula

I = (a·A + b·B + c·C) / (a + b + c)

Here a, b, c are the lengths of the sides opposite vertices A, B, C respectively (a=|BC|, b=|CA|, c=|AB|).

Inradius

r = S / s

S = triangle area, s = semi-perimeter = (a+b+c)/2

Position of the Incenter

The incenter is always inside the triangle. This distinguishes it from the circumcenter.

Alternative Expression

r = (s−a)·tan(A/2)

r = (s−b)·tan(B/2)

r = (s−c)·tan(C/2)

Worked examples

A(0, 0), B(4, 0) and C(0, 3) has sides 5, 3 and 4 opposite A, B and C, so the incenter is (1, 1) and the inradius 1.

An equilateral triangle with A(0, 0), B(6, 0) and C(3, 5.1962) puts the incenter at (3, 1.7321) with an inradius of the same 1.7321.

What the coordinates assume

Each vertex is weighted by the length of the side opposite it, not averaged equally — an equal average would give the centroid instead. The incenter always lies inside a real triangle, and collinear points produce an error. Coordinates, radius and distance share one unit.

Frequently asked questions

Why are the vertices weighted by the opposite sides?: The angle bisectors cut the opposite side in the ratio of the adjacent sides, and the weighted average is the coordinate form of that theorem. Averaging the vertices equally yields the centroid, a different point.
How does the inradius follow from the coordinates?: With r = area ÷ semi-perimeter. For the 3-4-5 triangle the area is 6 and the semi-perimeter 6, so r = 1, matching the distance from (1, 1) to either leg.
Does the incenter coincide with another center in an equilateral triangle?: Yes. In an equilateral triangle the incenter, centroid, circumcenter and orthocenter are the same point, one third of the height above the base.