Condition for Circles to be Orthogonal

Theorem
Let $\CC_1$ and $\CC_2$ be circles embedded in a Cartesian plane.

Let $\CC_1$ and $\CC_2$ be described by Equation of Circle in Cartesian Plane as:

Then $\CC_1$ and $\CC_2$ are orthogonal :
 * $2 \alpha_1 \alpha_2 + 2 \beta_1 \beta_2 = c_1 + c_2$

Proof
When $\CC_1$ and $\CC_2$ are orthogonal, the distance between their centers forms the hypotenuse of a right triangle whose legs are equal to the radii.

From Equation of Circle in Cartesian Plane: Formulation 3, the radii $R_1$ and $R_2$ of $\CC_1$ and $\CC_2$ respectively are given by:
 * $c_1 = \alpha_1^2 + \beta_1^2 - r_1^2$
 * $c_2 = \alpha_2^2 + \beta_2^2 - r_2^2$

Hence we have: