In the following construction you see a circle c with the equation x² + y² = 25, its center M = (0, 0) und the radius r = 5.

1. Double click on the circle's equation "c: x² + y² = 25" in the left window, change the right hand side of the equation and press Enter. What happens? Change the right hand side repeatedly and write down your observations.
2. Now, try to change the right hand side of the circle's equation in such a way that the radius becomes a) r = 2, b) r = 4 and c) r= 6. What does the equation look like for an abstract radius r? Write down your results and conjectures.
3. Drag the circle with the mouse and keep track of its equation and its midpoint. Is there any connection between them? Write down your observations and conjectures.
4. Use the keyboard to change the circle's equation so that its center becomes a) M = (4, 2), b) M = (3, -2) and c) M = -2, -1). Write down the resulting equations.
5. Do you have an idea what the equation of a circle with abstract center M = (m, n) and radius r could look like? Write down your conjectures.