Theorem: The angle of intersection of two lines and their inverse images

Statement

Let K be a circle and and be lines that intersect at point C.

The angle of intersection of and at point C is the same as the angle of intersection of their inverse images with respect to circle K.

Diagram

Proof

Let the circle be the inverse of .

Let the circle be the inverse of .

Both and pass through the center of circle K, point O.

Let be the foot of the perpendicular segment from O to line .

Let be the foot of the perpendicular segment from O to line .

Note that contains the center of , and contains the center of .

Let be the tangent to circle , and be the tangent to circle .

Then is perpendicular to , and is perpendicular to .

But, is also perpendicular to , so is parallel to .

Similarly is perpendicular to , so is parallel to .

So, angle APB, the angle of intersection of the two lines, is equal to angle A'OB', the angle of intersection of their inverse images with respect to K.

Q.E.D.

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