Incenter Calculator
Calculate the incenter and inradius of a triangle from vertex coordinates
Enter Vertex Coordinates
Incenter (I)
Inradius (r)
Side Lengths:
a (BC) = 0.000
b (AC) = 0.000
c (AB) = 0.000
Visualization
About the Incenter
The incenter is the center of the inscribed circle (incircle) of a triangle. It is the point where the three angle bisectors of the triangle intersect. The incenter is equidistant from all three sides of the triangle.
Key Properties
- The incenter is always located inside the triangle
- It is equidistant from all three sides of the triangle
- The distance from the incenter to any side is called the inradius (r)
- The incenter is the center of the largest circle that can fit inside the triangle
- The angle bisectors always meet at the incenter
Formulas
I = (a·A + b·B + c·C) / (a + b + c)
r = Area / s
Where a, b, c are side lengths, A, B, C are vertices, and s is the semiperimeter.
Frequently Asked Questions
What is the incenter of a triangle?
The incenter is the point where the three angle bisectors of a triangle intersect. It's the center of the inscribed circle (incircle) that touches all three sides of the triangle from the inside.
How do you find the incenter coordinates?
The incenter coordinates are calculated using a weighted average of the vertices, where the weights are the opposite side lengths. The formula is I = (a·A + b·B + c·C) / (a + b + c), where a, b, c are the side lengths.
What is the inradius?
The inradius is the radius of the inscribed circle. It's the distance from the incenter to any side of the triangle. It can be calculated using the formula r = Area/s, where s is the semiperimeter.
Is the incenter always inside the triangle?
Yes, the incenter is always located inside the triangle, regardless of whether the triangle is acute, right, or obtuse. This makes it different from the circumcenter and orthocenter, which can be outside.
What's the difference between incenter and centroid?
The incenter is the intersection of angle bisectors and the center of the inscribed circle, while the centroid is the intersection of medians and represents the center of mass. They are generally at different locations.
Why is the incenter important?
The incenter is important in geometry and practical applications. It's used in circle packing problems, computer graphics, engineering design, and architecture where you need to find the largest circle that fits inside a triangular space.
How is the inscribed circle related to the triangle's area?
The area of a triangle can be calculated as A = r·s, where r is the inradius and s is the semiperimeter. This relationship connects the incenter properties directly to the triangle's area.
Can two different triangles have the same incenter?
No, each triangle has a unique incenter determined by its three vertices. However, different triangles can have inscribed circles with the same radius (inradius) if they have the same area and semiperimeter.