Bisector Properties 623740

1. **State the problem:** We have a segment $CE$ with two bisectors: $OR$ which is a perpendicular bisector, and $ST$ which is another bisector. We want to conclude something about $ST$. 2. **Recall the definition of a perpendicular bisector:** A perpendicular bisector of a segment not only divides the segment into two equal parts but also forms a right angle (90 degrees) with the segment. 3. **Important property:** A segment can have only one unique perpendicular bisector. This is because the midpoint and the right angle condition uniquely determine the line. 4. **Since $OR$ is a perpendicular bisector of $CE$, it means:** - $OR$ passes through the midpoint of $CE$. - $OR$ is perpendicular to $CE$. 5. **What about $ST$?** Since $ST$ is also a bisector of $CE$, it must pass through the midpoint of $CE$. 6. **Conclusion:** Because the perpendicular bisector is unique, $ST$ cannot be a different perpendicular bisector. Therefore, $ST$ must be the same line as $OR$ or at least coincide with it. If $ST$ is not perpendicular, then it is just a bisector (passes through midpoint) but not perpendicular. 7. **Summary:** The second bisector $ST$ must pass through the midpoint of $CE$. If $ST$ is also a perpendicular bisector, then $ST$ coincides with $OR$. Otherwise, $ST$ is just a bisector (not perpendicular).