1. **Problem Statement:** Given a line segment $\overline{CD}$, construct its perpendicular bisector and label the intersection point as $E$. 2. **Construction of the Perpendicular Bisector:** - The perpendicular bisector of a segment is a line that divides the segment into two equal parts at a right angle (90 degrees). - To construct it, find the midpoint $E$ of $\overline{CD}$ such that $CE = ED$. - Draw a line through $E$ perpendicular to $\overline{CD}$. 3. **Relationship Between $m \overline{CE}$ and $m \overline{ED}$:** - Since $E$ is the midpoint of $\overline{CD}$, by definition, $m \overline{CE} = m \overline{ED}$. - This means the lengths of segments $CE$ and $ED$ are equal. 4. **Reasoning:** - The perpendicular bisector always passes through the midpoint of the segment it bisects. - Therefore, it divides $\overline{CD}$ into two equal parts. **Final answer:** $$m \overline{CE} = m \overline{ED}$$ This shows that $E$ is equidistant from $C$ and $D$ along the segment $\overline{CD}$.