IMO Shortlist 1968 problem 8


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2. travnja 2012.
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Given an oriented line \Delta and a fixed point A on it, consider all trapezoids ABCD one of whose bases AB lies on \Delta, in the positive direction. Let E,F be the midpoints of AB and CD respectively. Find the loci of vertices B,C,D of trapezoids that satisfy the following:

(i) |AB| \leq a (a fixed);

(ii) |EF| = l (l fixed);

(iii) the sum of squares of the nonparallel sides of the trapezoid is constant.

Remark
Remark. The constants are chosen so that such trapezoids exist.
Izvor: Međunarodna matematička olimpijada, shortlist 1968