Why does the equation x^2+y^2=r^2 form a circle in the Cartesian plane?

The general line equation in the Cartesian plane is given by ax+by=c, where a, b, and c are given constants. This means that all (x_0, y_0) points that are situated on the line satisfy the equation ax_0+by_0=c. In general, an equation of a line, a circle, an ellipse or other curves is an equation such that all points on the curve satisfy the equation of the curve and other points on the plane do not.
The equation x^2+y^2=r^2 has the equivalent form squareroot(x^2+y^2)=r. According to the distance formula in the Cartesian coordinate system, this means that all points on the curve are of equal distance from the origin. This is exactly the definition of the circle, namely that all points on the circumference are of equal distance from the center of the circle.