Equation of Circle/Cartesian/Formulation 1

Theorem

The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\tuple {a, b}$ can be expressed as:

$\paren {x - a}^2 + \paren {y - b}^2 = R^2$

Let the point $\tuple {x, y}$ satisfy the equation:

$(1): \quad \paren {x - a}^2 + \paren {y - b}^2 = R^2$

By the Distance Formula, the distance between this $\tuple {x, y}$ and $\tuple {a, b}$ is:

$\sqrt {\paren {x - a}^2 + \paren {y - b}^2}$

But from equation $(1)$, this quantity equals $R$.

Therefore the distance between points satisfying the equation and the center is constant and equal to the radius.

Thus $\tuple {x, y}$ lies on the circumference of a circle with radius $R$ and center $\tuple {a, b}$.

Now suppose that $\tuple {x, y}$ does not satisfy the equation:

$\paren {x - a}^2 + \paren {y - b}^2 = R^2$

Then by the same reasoning as above, the distance between $\tuple {x, y}$ and $\tuple {a, b}$ does not equal $R$.

Therefore $\tuple {x, y}$ does not lie on the circumference of a circle with radius $R$ and center $\tuple {a, b}$.

Hence it follows that the points satisfying $(1)$ are exactly those points which are the circle in question.

$\blacksquare$

1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $14$. To find the equation of the circle whose centre is $\tuple {\alpha, \beta}$ and radius $r$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 10$: Formulas from Plane Analytic Geometry: $10.15$: Equation of Circle of Radius $R$, Center are $\tuple {x_0, y_0}$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): circle
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): circle
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): circle
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): circle