Circle is Ellipse with Equal Major and Minor Axes

Theorem

Let $E$ be an ellipse whose major axis is equal to its minor axis.

Then $E$ is a circle.

Proof

Let $E$ be embedded in a Cartesian plane in reduced form.

Then from Equation of Ellipse in Reduced Form $E$ can be expressed using the equation:

$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$

where the major axis and minor axis are $a$ and $b$ respectively.

Let $a = b$.

Then:

 $\ds \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2}$ $=$ $\ds 1$ $\ds \leadsto \ \$ $\ds x^2 + y^2$ $=$ $\ds a^2$

which by Equation of Circle center Origin is the equation of a circle whose radius is $a$.

$\blacksquare$