I think the work and the ideas behind this are so good. You are able to get so much from this. Keep up the good work.
dmti

The standard eqn. of a circle is x^2+y^2+2gx+2fy+c=0
The coordinate of the center is (-g,-f)

Since 3 points lie on the circle , they must satisfy the eqn. of the circle . Then put the coordinates of those (3pts.) on the eqn. of the circle to obtain 3 eqns. in variables g,f &c . Solve them to get the values of g,f,&c. Hence the center (-g,-f ) .

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``Old theorems never die; they turn into definitions.''
----E. Hewitt

''ALPHA" =)