Triangle Centroid Calculator
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About Triangle Centroid
The centroid of a triangle is the point where all three medians of the triangle intersect. A median is a line segment that joins a vertex to the midpoint of the opposite side.
Properties of the Centroid
- The centroid divides each median in the ratio 2:1 (the distance from any vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side).
- The centroid is the center of mass of the triangle (assuming the triangle has uniform density).
- The centroid is located at the average of the coordinates of the three vertices.
Centroid Formula
For a triangle with vertices at $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the centroid $(x_G, y_G)$ is given by:
$x_G = \frac{x_1 + x_2 + x_3}{3}$
$y_G = \frac{y_1 + y_2 + y_3}{3}$
Applications
- Finding the center of mass of a triangular object
- Locating the balance point of a triangular plate
- Determining the center of a triangular region for various engineering and design applications
- Used in computational geometry and computer graphics
Other Triangle Centers
Besides the centroid, a triangle has several other important centers:
- Incenter: The center of the inscribed circle, located at the intersection of the angle bisectors.
- Circumcenter: The center of the circumscribed circle, located at the intersection of the perpendicular bisectors of the sides.
- Orthocenter: The intersection of the three altitudes of the triangle.
- Euler Line: The line passing through the centroid, orthocenter, and circumcenter.