Incenter, Incircle, and Angle Bisectors
The incenter of a triangle is the unique point where the three internal angle bisectors meet.
An angle bisector is a ray that splits an angle into two equal angles. In a triangle, the bisector from vertex \(A\) divides the
angle \(\angle A\) into two angles of equal measure, and similarly for vertices \(B\) and \(C\). A fundamental theorem of Euclidean
geometry says that these three internal bisectors always intersect at a single point—this intersection is the incenter, often denoted \(I\).
The incenter is important because it is equidistant from all three sides of the triangle. “Distance to a side” means
the shortest distance from a point to the line containing that side, which is measured along a perpendicular. Because all three distances
from \(I\) to the sides are the same, you can draw a circle centered at \(I\) with radius equal to that common distance, and the circle will
touch (be tangent to) each side. This circle is called the incircle, and its radius is the inradius \(r\).
The tangency points are where the perpendicular radii from \(I\) meet each side.
There is a clean relationship between the incircle, the triangle’s area, and its perimeter. Let the side lengths be \(a\), \(b\), and \(c\),
and define the semiperimeter \(s=\frac{a+b+c}{2}\). The area \(\Delta\) of the triangle can be expressed as the sum of the
areas of three smaller triangles that share the same height \(r\): one triangle has base \(a\) and height \(r\), another has base \(b\),
and the third has base \(c\). Adding those areas gives
\[
\Delta = \frac{1}{2}ar + \frac{1}{2}br + \frac{1}{2}cr = r\cdot\frac{a+b+c}{2} = rs.
\]
Solving for the inradius yields the formula used in this calculator:
\[
r = \frac{\Delta}{s}.
\]
When side lengths are known, \(\Delta\) can be found with Heron’s formula:
\[
\Delta = \sqrt{s(s-a)(s-b)(s-c)}.
\]
Once the triangle is constructed, the incenter can also be computed from coordinates. A particularly convenient expression is a
side-length–weighted average of the vertices:
\[
I = \frac{aA + bB + cC}{a+b+c},
\]
where \(a\) is the length opposite vertex \(A\), and similarly for \(b\) and \(c\). This reflects the geometric fact that the bisectors
“weight” the vertices according to the opposite side lengths.
The incenter always lies inside the triangle (unlike the circumcenter, which may lie outside for obtuse triangles).
If the triangle is degenerate—meaning the triangle inequality fails or the area is essentially zero—an incircle cannot be formed, so
this tool reports an error instead of producing misleading values.
