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5

Part of 1973 Czech and Slovak Olympiad III A

Problems(1)

Unknown geometric transformation

Source: Czech and Slovak Olympiad 1973, National Round, Problem 5

7/16/2024
Given two points P,QP,Q of the plane, denote P+QP+Q the midpoint of (possibly degenerate) segment PQPQ and P⋅QP\cdot Q the image of PP in rotation around the origin QQ under +90∘.+90^\circ. a) Are these operations commutative? b) Given two distinct points A,BA,B the equation Y⋅X=(A⋅X)+BY\cdot X=(A\cdot X)+B defines a map X↦Y.X\mapsto Y. Determine what the mapping is. c) Construct all fixed points of the map from b).
geometrygeometric transformationrotation