1. **Problem Statement:** We are given triangle $ABC$ with $AE = EC$, $m\angle BAC = m\angle DFE$, $AC = 24$, and $m\angle DFE = 126^\circ$. We need to find the length of segment $DF$. 2. **Understanding the Problem:** Since $AE = EC$, point $E$ is the midpoint of segment $AC$. This means $AE = EC = \frac{AC}{2} = \frac{24}{2} = 12$. 3. **Key Properties:** The equality of angles $m\angle BAC = m\angle DFE$ and the given $m\angle DFE = 126^\circ$ imply $m\angle BAC = 126^\circ$. 4. **Using the Midsegment Theorem:** In triangle $ABC$, segment $DE$ is a midsegment if $D$ and $E$ are midpoints of sides $AB$ and $AC$ respectively. The midsegment theorem states that $DE$ is parallel to $BC$ and $DE = \frac{1}{2} BC$. 5. **Finding $DF$:** Since $D$ and $E$ are midpoints, and $F$ lies on $BC$, $DF$ is half the length of $BC$ if $F$ is the midpoint of $BC$. However, the problem does not explicitly state $F$ is midpoint, but given the angle conditions and the parallelogram-like figure, $DF$ corresponds to the length of the midsegment. 6. **Calculate $DF$:** Since $AC = 24$ and $E$ is midpoint, $AE = EC = 12$. The triangle's side $BC$ is unknown, but by the properties of the midsegment and the given angles, $DF$ equals $12$. **Final answer:** $$DF = 12$$