Polar Bears. Polar Bears like the cold. Some live at the North Pole. Some live in Canada and Alaska.
Pole Polar
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Pole-Polar: Key Facts Tarik 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). Let x and y be the polars of X and Y , respectively. Then X is on line y ⇐⇒ Y is on line x. Proof. Let X , Y be the images of X, Y for the inversion with respect to C . Then OX.OX = r 2 = OY.OY = ⇒ X.X ,Y,Y are concyclic. Now, X ∈ y ⇐⇒ ∠X Y Y = 90 o ⇐⇒ ∠X X Y = 90 o ⇐⇒ Y ∈ x X' O X Y x Y' Theorem 1.2. Let x, y,z be the polars of distinct points X,Y,Z resp ectively. Then Z = x ∩ y ⇐⇒ z = X Y Proof. By La Hire’s theorem, Z on x ∩ y ⇐⇒ X on z and Y on z ⇐ ⇒ z = X Y . z O X Y Z 1
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