Loci: unexpected family of curves
- Juan Carlos Ponce Campuzano
- Consider a line f and a point C defined on that line. Let's draw a circle c with centre C and radius CR.
- Consider a point B on f, and point A on the circle. Reflect point A with respect to line f to determine point A'.
- 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?
- 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.
In the following applet, select the tool Locus 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'?
If B is moved, the locus of D (with respect of A) describes a family of curves. Can you name them?