By definition, a parabola is the set of points equidistant from a line, called the directrix, and a fixed point, called the focus. Assuming the focus is not on the line, here's the construction of a parabola given a fixed point for the focus and a line (segment) for the directrix.

1) Construct a line segment called the directix and a fixed point not on the directrix called the focus.

2) Construct a point on the directrix where you can move the point along the line segment.

3) Construct a perpendicular line to the directrix on the moving point.

4) Construct a segment using the moving point on the directrix and the focus.

5) Construct a midpoint from the above segment.

6) Construct a perpendicular line from the segment on the midpoint.

7) The intersection of two perpendicular lines is called the locus.

Here's a picture constructing a parabola locus using GSP.

Here's a picture where the locus is being traced.

Click here for the actual GSP file.

When you trace the tangent line, here's the picture.

Click here for GSP file.

Use the locus command to generated the parabola from a constructed point or the tangent line at that point.

Here's a picture when I use the locus command from GSP.

Click here for the GSP file.

Now, I'm going to repeat the steps of constructing a parabola locus, but this time, I'm going to use a circle instead of a line (segment) for the directrix.

What if the focus is inside the circle?

The locus became an ellipse!

What if the focus is outside the circle?

Then it creates a hyperbola locus.

Click here for a GSP investigation.

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