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Prove the existence of this triangle.

Source: Costarrican Math Olympiad 2006 - Problem 3.

May 20, 2006
vectorgeometryparallelogramgeometry unsolved

Problem Statement

Let ABCABC be a triangle. Let P,Q,RP, Q, R be the midpoints of BC,CA,ABBC, CA, AB respectively. Let U,V,WU, V, W be the midpoints of QR,RP,PQQR, RP, PQ respectively. Let x=AU,y=BV,z=CWx=AU, y=BV, z=CW. Prove that there exist a triangle with sides x,y,zx, y, z.
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