Enter the coordinates of all three vertices — A(x, y), B(x, y), and C(x, y) — of your triangle, and the Triangle Incenter Calculator returns the incenter coordinates (Ix, Iy) and the inradius. The incenter is the point where all three angle bisectors meet and serves as the center of the triangle's inscribed circle.
Incenter X (Ix)
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Incenter Y (Iy)
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Inradius (r)
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Side a (BC)
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Side b (CA)
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Side c (AB)
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Perimeter
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Triangle Area
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The incenter is the point where all three interior angle bisectors of a triangle intersect. It is equidistant from all three sides of the triangle, making it the center of the triangle's inscribed circle (incircle). Unlike some other triangle centers, the incenter always lies inside the triangle.
The incenter coordinates are a weighted average of the three vertex coordinates, where the weights are the lengths of the opposite sides. The formula is: Ix = (a·Ax + b·Bx + c·Cx) / (a + b + c) and Iy = (a·Ay + b·By + c·Cy) / (a + b + c), where a, b, c are the lengths of the sides opposite to vertices A, B, and C respectively.
The inradius (r) is the radius of the largest circle that fits entirely inside the triangle, touching all three sides. It is calculated as r = Area / s, where Area is the triangle's area and s is the semi-perimeter (half the perimeter). A larger inradius indicates a more 'rounded' triangle.
No — unlike the circumcenter or orthocenter, the incenter always lies strictly inside the triangle, regardless of whether the triangle is acute, right, or obtuse. This is because angle bisectors of interior angles always converge inside the polygon.
The incenter is the center of the inscribed circle (incircle) that touches all three sides, found at the intersection of angle bisectors. The circumcenter is the center of the circumscribed circle (circumcircle) that passes through all three vertices, found at the intersection of perpendicular bisectors of the sides. The circumcenter can lie outside the triangle for obtuse triangles.
If all three vertices lie on a straight line, they do not form a valid triangle — the area is zero and the perimeter-based formula breaks down (division by zero for the inradius). The calculator handles this by checking that the computed area is non-zero before displaying results.
Translating (shifting) the triangle moves the incenter by the same translation. Scaling the triangle scales the incenter's distance from the origin but keeps it at the same relative position within the triangle. The incenter's position relative to the triangle's shape is a geometric property and does not change with uniform scaling.
Yes. The incenter formula works for any real-valued coordinates, including negative values. Simply enter the correct x and y coordinates for each vertex, and the calculator will compute the correct incenter and inradius regardless of the coordinate signs or quadrant placement.