 Circles. By Pythagoras’s Theorem the square of the distance between two points \((x, y)\) and \((x', y')\) is given by \((x'-x)^2 +(y'-y)^2\). As a special case of this, the square of the distance of the point \((x, y)\) to the origin is \(x^2 +y^2\). It follows that those point at distance \(r\) from the origin are the solutions in \(x\) and \(y\) to the equation \(x^2 +y^2 = r^2\). But these points are exactly those forming the circle of radius \(r\) centered on \(O\). We identify this equation with this circle.