Triangle Explorations

by

Laura Kimbel

Construct the locus of points equidistant from a fixed point F and a circle. This is similar to a parabola construction but I will use a circle as the "directrix." Let F be any point in the plane other than the center of the circle. Assume F is not on the circle; it can be either inside or outside.

To make this construction, first draw a circle and a fixed point, and locate 1 additional point on the circle.

Next, make a line segment between your fixed point and the point on the circle.

Finally, locate the midpoint of your segment, and construct a perpendicular bisector. Construct the locus by selecting the midpoint and the point on the circle. The final construction should look like this:

So, the locus of points equidistant from a fixed point and a circle is another circle.

What can we tell about the radii of each of the circles? Through the construction of similar triangles, we can see that the radius of the locus is 1/2 the radius of our circle.

First, I drew a line segment from our fixed point to the center of the green circle. Next, I constructed a line parallel to the radius of our green circle but through our midpoint. The intersection of these 2 line segments is the center of our circle created by the locus. If we analyze these 2 triangles, we can see that they are similar, and in fact, each side of the small triangle is 1/2 the length of the corresponding side of the larger triangle. This means the radius of the red circle is 1/2 the radius of the green circle. We can see that this is true for all locations of our fixed point, except when out fixed point is placed at the center of our green circle. CLICK HERE to observe different cases.

What if we look at the locus of points equidistant from 2 circles?

If you let the points rotate for long enough, tracing the midpoint, you will see that a "washer" is produced. However, you will also notice that the washer is produced with different paths for each button.

The washer has some interesting properties. The outer radius is

(R1+R1)/2.

|R1-R2|/2.

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