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The value of $x$, which is $\angle AGB$, is $\mathbf{120^\circ}$.\r
\n" ); document.write( "\n" ); document.write( "Here is the reasoning based on the properties of an equilateral triangle and its centroid.\r
\n" ); document.write( "\n" ); document.write( "---\r
\n" ); document.write( "\n" ); document.write( "## Centroid and Medians in an Equilateral Triangle\r
\n" ); document.write( "\n" ); document.write( "1. **Medians are also Angle Bisectors and Altitudes:** In an **equilateral triangle** ($\triangle ABC$), the medians (which intersect at the centroid $G$) are also the angle bisectors and the altitudes.
\n" ); document.write( "2. **Symmetry:** Because $\triangle ABC$ is equilateral, it has three-fold rotational symmetry. This means that the area, angles, and distances formed by the centroid to the three vertices are identical.
\n" ); document.write( "3. **Angles around the Centroid:** The centroid $G$ divides the triangle into three smaller triangles: $\triangle AGB$, $\triangle BGC$, and $\triangle CGA$. The angles formed at the centroid ($\angle AGB$, $\angle BGC$, and $\angle CGA$) must sum to $360^\circ$.
\n" ); document.write( " $$\angle AGB + \angle BGC + \angle CGA = 360^\circ$$
\n" ); document.write( "4. **Equal Angles:** Due to the symmetry of the equilateral triangle, these three central angles must be equal:
\n" ); document.write( " $$\angle AGB = \angle BGC = \angle CGA$$\r
\n" ); document.write( "\n" ); document.write( "### Calculation\r
\n" ); document.write( "\n" ); document.write( "Let $x = \angle AGB$.
\n" ); document.write( "$$x + x + x = 360^\circ$$
\n" ); document.write( "$$3x = 360^\circ$$
\n" ); document.write( "$$x = \frac{360^\circ}{3} = \mathbf{120^\circ}$$\r
\n" ); document.write( "\n" ); document.write( "Therefore, $\angle AGB = 120^\circ$. \n" ); document.write( "