Centroid of a Triangle
The centroid of a triangle is the point where its three medians intersect.
A median is the segment drawn from a vertex to the midpoint of the opposite side. If a triangle has vertices
\(A\), \(B\), and \(C\), then the median from \(A\) goes to the midpoint \(D\) of side \(BC\), the median from \(B\)
goes to the midpoint \(E\) of side \(CA\), and the median from \(C\) goes to the midpoint \(F\) of side \(AB\).
The medians are special because they always meet at exactly one point, and that point is the centroid \(G\).
A key property is the 2:1 ratio rule: the centroid divides each median so that the portion from the
vertex to the centroid is twice the portion from the centroid to the midpoint. In symbols,
\[
AG:GD = 2:1,\quad BG:GE = 2:1,\quad CG:GF = 2:1.
\]
Equivalently, \(AG=\tfrac{2}{3}m_a\) and \(GD=\tfrac{1}{3}m_a\), where \(m_a\) is the length of the median from \(A\).
The same pattern holds for the other two medians. This is why the centroid is often described as the triangle’s
“balance point”: if the triangle were a thin uniform plate, it would balance perfectly on a pin placed at \(G\).
When you do not start with coordinates, you can still compute median lengths from the three side lengths
\(a=|BC|\), \(b=|CA|\), and \(c=|AB|\). The classic result is Apollonius’ theorem, which leads to the formulas
\[
m_a=\tfrac12\sqrt{2b^2+2c^2-a^2},\quad
m_b=\tfrac12\sqrt{2a^2+2c^2-b^2},\quad
m_c=\tfrac12\sqrt{2a^2+2b^2-c^2}.
\]
These expressions work for any valid triangle, including acute, right, and obtuse triangles. Once you know the
medians, the centroid distances follow immediately using the \(\tfrac{2}{3}\) and \(\tfrac{1}{3}\) factors.
This calculator focuses on non-coordinate input (sides or medians), but it still provides a visual preview.
For the plot only, the tool constructs one triangle that matches your side lengths and then draws midpoints,
medians, and the centroid. The computed results (median lengths and centroid ratios) do not depend on that specific
drawing; they are geometric facts that hold for every triangle with the same side lengths.
Mini glossary
- Median: segment from a vertex to the midpoint of the opposite side.
- Centroid: intersection point of the three medians; the triangle’s center of mass for uniform density.
- 2:1 ratio: vertex-to-centroid is twice centroid-to-midpoint along each median.
