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Another hexagon geometry -- angles on midpoints of sides

Source: Taiwan 2014 TST3 Quiz 2, P1

July 18, 2014
geometryvectorgeometry proposed

Problem Statement

In convex hexagon ABCDEFABCDEF, AB∥DEAB \parallel DE, BC∥EFBC \parallel EF, CD∥FACD \parallel FA, and AB+DE=BC+EF=CD+FA. AB+DE = BC+EF = CD+FA. The midpoints of sides ABAB, BCBC, DEDE, EFEF are A1A_1, B1B_1, D1D_1, E1E_1, and segments A1D1A_1D_1 and B1E1B_1E_1 meet at OO. Prove that ∠D1OE1=12∠DEF\angle D_1OE_1 = \frac12 \angle DEF.
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