Construction of the Locus of Points Equidistant from a Circle and a Fixed Point

Once you have your circle and focus on the page, place a point on the circle (I have labeled it point A). Select point A and the focus F and use the Construct menu to construct the segment between A and F. Goto the Construct menu again to find the midpoint of segment AF (labeled M). Select point M and segment AF. Go once again to the Construct menu and select Perpendicular Line. Your sketch should look like this so far.

This is where we have to be a little clever with our new sketch. Remember, we are trying to find the point (or locus of points really) that is equidistant from F and A. If we can create an isosceles triangle with base AF, we will be on the mark. The reason for constructing the perpendicular bisector through AF is that any point on this bisector will be the same distance from A as it is from F. To construct such a point, we can Construct a line using the center of our circle and point A as two points that define that line. We will Construct a point (labeled L for locus) at the intersection of the two lines.

The following sketch clearly shows our isosceles triangle.

Now we are ready to explore with our sketch.