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Midpoint is a Fixed Point - [Iran Second Round 1986]

Source:

December 24, 2010
rotationgeometrygeometric transformationreflectionsymmetrygeometry proposed

Problem Statement

OO is a point in the plane. Let O′O' be an arbitrary point on the axis OxOx of the plane and let MM be an arbitrary point. Rotate MM, 90∘90^\circ clockwise around OO to get the point M′M' and rotate MM, 90∘90^\circ anticlockwise around O′O' to get the point M′′.M''. Prove that the midpoint of the segment MM′′MM'' is a fixed point.
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