The knowledge of which geometry aims is the knowledge of the eternal. — Plato
2L solution
Let O be the point.
- Construct a circle with center O and an arbitrary radius so that it intersects the line at two points A and B
- Construct the perpendicular bisector of AB
3E solution
Let O be the point.
- Construct a circle with an arbitrary center A on the line and radius OA
- Construct a circle with center B at intersection of circle A and the line and radius OB, intersecting circle A at C
- Construct line OC
Explanation
Solution 2L has you construct an arbitrary isosceles triangle ABO with O as its apex. The perpendicular bisector of its base will pass through its apex and therefore will answer the puzzle.
Solution 3E has you construct the reflection of point O through the line, making that line the perpendicular bisector of the segment joining point O to its reflection. Just trace that segment.