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Loci: unexpected family of curves

Part I

Geometric context:
  1. Consider a line f and a point C defined on that line. Let's draw a circle c with centre C and radius CR.
  2. Consider a point B on f, and point A on the circle. Reflect point A with respect to line f to determine point A'.
  3. Finally, we draw the lines AB and A'C; these lines intersect at D. What is the locus of point D when point A is moved across the circle?
Explore the following in the applet:
  • Move the point A and pay attention to point D. Click Animate A for automatic motion.
  • Activate the trace of D to help you to observe a pattern.
  • Change the position of B along the line f, and move A again.

Part II

In the following applet, select the tool Locus Toolbar Image and apply it to the point D when A is moved. Then, move only the point B and observe what happens to the locus of D. Click on Animate B for automatic motion. Questions:
  • What happens if B is within the segment EE'?
  • What happens if B is not in the segment EE'?
  • What happens if B is equal to E or E'?
Note: The segment EE' is equal to 4CR.

Review

If B is moved, the locus of D (with respect of A) describes a family of curves. Can you name them?