Equation of a Circle

The equation of a circle can be derived using Pythagoras Theorem. The circle below has centre  
\[(x_1,y_1)\]
. The point  
\[(x,y)\]
  lies on the circle.

Construct a right angled triangle with hypotenuse from the centre  
\[(x_1,y_2)\]
  to the point  
\[(x,y)\]
  - and so equal to the radius  
\[r\]
  of the circle, base parallel to the  
\[x\]
  - axis and height parallel to the  
\[y\]
  - axis.
Then the base of the triangle is  
\[x-x_1\]
  and the height is  
\[y-y_1\]
.
From Pythagoras Theorem,  
\[(x-x_1)^2+(y-y_1)^2=r^2\]

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