Find the centroid, incenter, circumcenter, and orthocenter of a triangle from three vertex coordinates. Computes area, perimeter, side lengths, and angles.
| Center | Formula | Notes |
|---|---|---|
| Centroid | ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) | Intersection of medians; always inside the triangle |
| Incenter | (a·x₁+b·x₂+c·x₃)/(a+b+c), same for y | Intersection of angle bisectors; always inside |
| Circumcenter | Equidistant from all 3 vertices | Intersection of perpendicular bisectors; may be outside |
| Orthocenter | Intersection of altitudes | Inside for acute, on vertex for right, outside for obtuse |
| Nine-point center | Midpoint of circumcenter & orthocenter | Center of the nine-point circle |
The centroid of a triangle is the point where its three medians intersect. A median connects a vertex to the midpoint of the opposite side, and the centroid divides each median in a 2:1 ratio from the vertex. For a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃), the centroid is simply ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) — the arithmetic mean of the coordinates.
The centroid is also the center of mass of a uniform triangular plate, which is why physicists and engineers care about it: if you balanced a triangular sheet on a pin at its centroid, it would stay level. It is always located inside the triangle, regardless of the triangle's shape.
Beyond the centroid, every triangle has three other classic centers. The incenter — intersection of angle bisectors — is the center of the inscribed circle (incircle). The circumcenter — intersection of perpendicular bisectors — is the center of the circumscribed circle and may lie outside the triangle for obtuse triangles. The orthocenter — intersection of altitudes — lies inside acute triangles, at the right-angle vertex for right triangles, and outside for obtuse triangles. All four centers are collinear on the Euler line (except the incenter, which is generally off it).
This calculator takes three vertex coordinates and computes all four triangle centers, plus area, perimeter, side lengths, and interior angles. Presets let you explore equilateral, isosceles, right, and scalene cases. A reference table summarises each center's properties.
Use this page when you need the centroid together with the other triangle centers and coordinate-geometry checks. It keeps the vertex data, centroid formula, area, and related center relationships together so you can compare where the triangle's notable points sit in the same coordinate frame.
Centroid: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
Incenter: I = (a·x₁+b·x₂+c·x₃)/(a+b+c), same for y (weighted by opposite side lengths)
Circumcenter: equidistant from all 3 vertices (perpendicular bisector intersection)
Orthocenter: H = 3G − 2O (from the Euler line relation)
Area: ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|Result: Centroid from the three triangle vertices
The centroid is found by averaging the three x-coordinates and the three y-coordinates. The rest of the output then compares that centroid with other triangle centers and the area from the same vertex data.
Find the centroid, incenter, circumcenter, and orthocenter of a triangle from three vertex coordinates. Computes area, perimeter, side lengths, and angles. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
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The centroid is the intersection of the three medians. It is the center of mass of a uniform triangular plate and is always located inside the triangle at ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).
The centroid is the intersection of medians (center of mass), while the incenter is the intersection of angle bisectors (center of the inscribed circle). They coincide only for equilateral triangles.
Yes. For obtuse triangles, the circumcenter lies outside the triangle, on the opposite side of the longest side from the obtuse angle.
The Euler line passes through the centroid, circumcenter, orthocenter, and nine-point center. The centroid divides the segment from circumcenter to orthocenter in a 2:1 ratio.
The circumcenter is at the midpoint of the hypotenuse, and the orthocenter is at the vertex of the right angle.
The inradius r = Area / s, where s is the semi-perimeter (a+b+c)/2. The incenter is the center of this inscribed circle.
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