Theorem: (Orthogonal Circle Inversion)

If K and C are orthogonal circles, then the image of C is unchanged when inverted through circle K.

Proof:

Let circles K and C be orthogonal circles with center S in circle K.

Let T be a point of intersection of circles K and C.

Construct a line l through S that intersects circle K at T₁ and intersects circle C at points P and P₁.

Since STis the radius of circle K, then ST²= SP*SP₁by the Power Tangent Formula.

ST=ST₁ because both are radii of circle K.

By substitution, ST₁²=SP*SP₁.

Therefore, by the definition of inversion, P₁ is the inverted image of P and the image of circle C is unchanged when inverted through circle K.