CIRCLES
-----Warren Smith---------

The equation of a circle with radius R and center coordinates
(A,B)  is

        2        2    2
   (X-A)  + (Y-B)  = R  .

This is really just Pythagoras's theorem in action - the equation
is saying that the distance from (X,Y) to (A,B) is R.
That is because every point (X,Y) on the perimeter of the circle
is at distance R from its centerpoint.

Also, note, by expanding out the

        2       2            2
   (X-A)    =  X  - 2 A X + A

and/or expanding out the square of  (Y-B)  also, we can get
different forms of the same equation for the same circle.
You should feel able to interconvert between these different
forms of the same equation.

Finally, suppose we want to find the equation of
the circle which passes through 3 different points,
call them (X1, Y1)    (X2, Y2)   and  (X3, Y3).
Then we need to find the magic A,B, and R so that

         2         2    2
   (X1-A)  + (Y1-B)  = R  ,

         2         2    2
   (X2-A)  + (Y2-B)  = R  ,    and

         2         2    2
   (X3-A)  + (Y3-B)  = R  .

This is 3 equations in 3 unknowns A, B and R.  But
if you expand out the equations and subtract the 3rd equation
from the first 2 equations, then you get 2 equations in
only TWO unknowns  (since the subtraction causes the R's to go away
via cancellation)  namely A and B.  Furthermore these two equations
are LINEAR in A and in B.  So you know how to solve them.
Then with A and B known,  R is easy to find by simply computing
the left side of any one of our 3 original equations
(since now all of the quantities in all these left
hand sides are known) to get the square of R, then take
the square root.

done.