What is the resultant of forces acting from the center of an…

What is the resultant of forces acting from the center of an equilateral triangle, oriented toward the vertices, with equal magnitudes?

Step 1: Setup

Let $ABC$ be the equilateral triangle with centre (centroid) $G$. Three forces of equal magnitude $F$ act from $G$ towards the vertices. For an equilateral triangle the centre is equidistant from the vertices; call this circumradius $r$, so $|\overrightarrow{GA}| = |\overrightarrow{GB}| = |\overrightarrow{GC}| = r$ and

\[ \vec{F}_A = \frac{F}{r}\,\overrightarrow{GA}, \qquad \vec{F}_B = \frac{F}{r}\,\overrightarrow{GB}, \qquad \vec{F}_C = \frac{F}{r}\,\overrightarrow{GC}. \]

Step 2: Coordinates

Place $G$ at the origin; the vertices lie at angles $90^\circ$, $210^\circ$, $330^\circ$: \[A = r(0,\,1), \qquad B = r\left(-\tfrac{\sqrt{3}}{2},\,-\tfrac{1}{2}\right), \qquad C = r\left(\tfrac{\sqrt{3}}{2},\,-\tfrac{1}{2}\right),\] so the forces are

\[ \vec{F}_A = F(0,\,1), \qquad \vec{F}_B = F\left(-\tfrac{\sqrt{3}}{2},\,-\tfrac{1}{2}\right), \qquad \vec{F}_C = F\left(\tfrac{\sqrt{3}}{2},\,-\tfrac{1}{2}\right). \]

Step 3: Resultant

\[ R_x = F\left(0 - \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}\right) = 0, \qquad R_y = F\left(1 - \frac{1}{2} - \frac{1}{2}\right) = 0 \]

\[\vec{R} = \vec{F}_A + \vec{F}_B + \vec{F}_C = \vec{0}\]

The resultant is zero — the three forces are in equilibrium.

Step 4: General vector proof

For any triangle the centroid $G = \dfrac{A+B+C}{3}$ satisfies

\[ \overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC} = (A+B+C)-3G = \vec{0}. \]

In an equilateral triangle all three segments have the same length $r$, so the equal forces are the same multiple $F/r$ of these vectors: \[\vec{F}_A+\vec{F}_B+\vec{F}_C = \frac{F}{r}\left(\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}\right) = \vec{0}. \qquad \blacksquare\]

Step 5: Interpretation

The forces are $120^\circ$ apart, so the configuration looks the same after a $120^\circ$ rotation; a non-zero resultant would have to point in a preferred direction, which cannot exist. Equivalently, by the parallelogram rule $\vec F_A+\vec F_B$ has magnitude $F$ and points exactly opposite to $\vec F_C$: