Problem
GEO-B2-M05-P002 Points on the Circle of Inversion
#2
★★☆☆☆ Level 2 of 5
Prove that every point of the circle \(OP=R\) remains fixed under the inversion with centre \(O\) and radius \(R\).
Substitute \(OP=R\) into the definition of inversion.
For the image \(P'\), \(OP\cdot OP'=R^2\). If \(OP=R\), then \(R\cdot OP'=R^2\), so \(OP'=R=OP\). Point \(P'\) lies on ray \(OP\), hence \(P'=P\).
Students should internalise that the circle of inversion is fixed pointwise.