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(Text from Circles Are Awesome)
Circles Are Awesome
by Jeff Weir
Circles have a rather intersting property. Just like triangles, you only need three points to create one.
So how do three points "define" a circle? Well, let's start by drawing three points and calling them A, B and C.
Now that we have three points, let's draw two lines: one from A to B, and one from B to C.
Next, we'll find the midpoints of the line segments we just drew.
At each midpoint, draw a perpendicular line in both directions. We'll draw a point where the lines meet and call it D.
D is now the center of the circle. For proof, we'll draw a line from D to the points A, B and C.
These lines are all the same length. If we rotate them around point D, we'll find a circle has been created.
But will A, B and C always define a circle? Try moving the points around to find out.
You found the point where A, B and C don't define a circle.
When A, B and C make a perfectly straight line, the perpendicular lines will never intersect. This means the center of the circle doesn't exist.
So three points don't always define a circle.
Now that we know how to make a circle with three points, what happens when we use B, C and D to define E?
We get another circle. In fact, we can go on forever making circles from other circles. But that's for you to play with...