# Loci: unexpected family of curves

- Author:
- Juan Carlos Ponce Campuzano

## Part I

Geometric context:

- 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.

## Part II

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'**?

**EE'**is equal to 4**CR**.## Review

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