Given an arbitrary parabola, how would you find its axis using only a compass and (unmarked) straightedge?

Pick points P1 and P2 on opposite side of the vertex of the parabola.  Carefully draw the tangents to the parabola at points P1 and P1 and extend them until they cross; label this point Q1.  Using the compass measure the distance from P1 to Q1, and construct a perpendicular to the tangent at P2 of this length.  Measure the distance from P2 to Q1 and construct a perpendicular to the tangent at P1 of this length.  These will meet at a point we can label Q2, and Q1-->Q2 will be parallel to the axis of symmetry of the parabola.  (Based on the properties of tangents of a parabola).  Draw a perpendicular to Q1-Q2 at any point and extend in both directions to intersect the parabola at R1, R2.  Bisect R1-->R2 and draw a parallel line (can be done with compass and straight edge) to Q1-->Q2 through the bisect point.  Then new line is the axis of symmetry of the parabola

Posted by Kenny M on 2019-09-02 20:30:13