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Line given by rotation and homothety

Source: Czech and Slovak Olympiad 1970, National Round, Problem 5

July 4, 2024
geometrygeometric transformationrotationhomothetyLocusLine

Problem Statement

Let a real number kk and points S,A,SA=1S,A,SA=1 in plane be given. Denote A′A' the image of AA under rotation by an oriented angle φ\varphi with respect to center SS. Similarly, let A′′A'' be the image of A′A' under homothety with the factor 1cos⁡φ−ksin⁡φ\frac{1}{\cos\varphi-k\sin\varphi} with respect to center S.S. Denote the locus ℓ={A′′∣φ∈(−π,π],cos⁡φ−ksin⁡φ≠0}.\ell=\bigl\{A''\mid\varphi\in(-\pi,\pi],\cos\varphi-k\sin\varphi\neq0\bigr\}. Show that ℓ\ell is a line containing A.A.
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