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2 XY <= AD + BC, 3D geo inequality , X, Y midpoints of AB,CD

Source: 2017 (-18) Swedish Mathematical Competition p3

August 30, 2020
geometric inequalitygeometry3D geometrymidpoints

Problem Statement

Given the segments ABAB and CDCD not necessarily on the same plane. Point XX is the midpoint of the segment ABAB, and the point YY is the midpoint of CDCD. Given that point XX is not on line CDCD, and that point YY is not on line ABAB, prove that 2∣XY∣≤∣AD∣+∣BC∣2 | XY | \le | AD | + | BC |. When is equality achieved?
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