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This is a straightforward geometry problem involving an **equilateral triangle** and its **incenter**. The required length, $DE$, is simply the length of the segment connecting the midpoints of two sides of the triangle.\r
\n" ); document.write( "\n" ); document.write( "## 📐 Finding the Length of DE\r
\n" ); document.write( "\n" ); document.write( "Since $D$ is the **midpoint** of side $AB$, and $E$ is the **midpoint** of side $AC$, the segment $DE$ is the **midsegment** (or midline) of $\triangle ABC$.\r
\n" ); document.write( "\n" ); document.write( "### The Triangle Midsegment Theorem\r
\n" ); document.write( "\n" ); document.write( "The Midsegment Theorem states that the segment connecting the midpoints of two sides of a triangle is **parallel** to the third side and is **half the length** of the third side.\r
\n" ); document.write( "\n" ); document.write( "1. **Identify the third side:** The third side of $\triangle ABC$ that is parallel to $DE$ is $BC$.
\n" ); document.write( "2. **Determine the length of $BC$:** Since $\triangle ABC$ is **equilateral** with a side length of $4$, all sides have a length of $4$. Thus, $BC = 4$.
\n" ); document.write( "3. **Calculate $DE$:** The length of the midsegment $DE$ is half the length of $BC$.
\n" ); document.write( " $$DE = \frac{1}{2} BC = \frac{1}{2} (4) = \mathbf{2}$$\r
\n" ); document.write( "\n" ); document.write( "The length of $DE$ is **2**.\r
\n" ); document.write( "\n" ); document.write( "---\r
\n" ); document.write( "\n" ); document.write( "## 💡 Note on the Incenter (I)\r
\n" ); document.write( "\n" ); document.write( "The information that $I$ is the incenter of $\triangle ABC$ is **extraneous** (unnecessary) for finding the length of $DE$. The position of the midsegment $DE$ depends only on the midpoints $D$ and $E$, not on the location of the triangle's incenter or other special points. \n" ); document.write( "