Pole-Polar: Key FactsTarik Adnan Moon In this article, we shall provide key facts of Pole-Polar transformation, however without proof. Interested readers may find the proofs in v ariou s sourc es, referre d sources are certainl y good places to start. The main goal of this article is to provide a tool kit for problem solver. 1 Ba si c F ac ts Theorem 1.1 (La Hire’s Theorem). Letx andy be the polars ofXandY, respectively. ThenXis on line y ⇐⇒ Yis on line x. Proof. Let X, Ybe the images ofX, Yfor the inversion with respect to C. Then OX.OX=r 2 =OY.OY= ⇒ X.X,Y,Y are concyclic. Now, X∈ y ⇐⇒ ∠X YY= 90 o ⇐⇒ ∠X XY= 90 o ⇐⇒ Y∈ x X' O XYx Y' Theorem 1.2. Letx, y,z be the polars of distinct pointsX,Y,Z resp ectively. ThenZ=x ∩ y ⇐⇒ z = X YProof. By La Hire’s theorem, Zon x ∩ y ⇐⇒ Xon z and Yon z ⇐ ⇒ z = X Y. z O XYZ 1
