Geometric Construction of Parabola Vertex from 3 points and a Horizontal Axis

I was wondering how easy it would be to draw the correct parabola on a whiteboard given only 3 points (given that the parabola opens in the vertical direction). It's not as easy as connecting two points...it's not even that practical in the end. But it was fun figuring it out.

This can be accomplished as follows (provided that no points are vertically aligned):

1. draw 3 vertical lines through the points and label the one whose line is between the other two as B
2. label the point whose vertical line is closer to B as A and the other one C
3. also label the point that is to the left of B as L and the one to the right as R
4. locate the midpoints of LB and RB as L' and R'
5. draw a horizontal line through B and locate the intersection between the vertical through R and this horizontal line as U
6. on the vertical through L' mark the point that is as far above/below BU as L is below/above BU and label that point D
7. draw a vertical line that is as far to the right of B as L is to the left of B and label the intersection of that line and RB as e
8. make a point on the vertical through R' that is the same height as e and label that point E
9. Mark as S the intersection between line DE and BU. The vertical through S is the axis of symmetry of the parabola.
10. make points the same height as A and B on the opposite side of the vertical through S and label them A' and B'

From here, credit to  http://mathafou.free.fr/themes_en/parabole2.html for the method of now locating the vertex.

1. mark where line AB' crosses the vertical through S as O
2. mark where line UO crosses the vertical through A as U'
3. draw a line from U' to B; this is the tangent to the parabola where it passes through B
4. reflect the vertical through B from U'B and where it cross the vertical through S is the focal point, F
5. the directrix is as far below B as the distance from B to F, mark that point below B as x
6. mark the point where the horizontal through x crosses the vertical through S as X
7. the vertex of the parabola is the midpoint between F and X, V.

It would look better in an animated GIF but here is an example with points and lines marked: