We can apply the cosine rule to triangles AOD and AOB:
AD^2=AO^2+OD^2-2AO*ODcosAOD
AB^2=AO^2+OB^2-2AO*OBcosAOB.
But AD=BC, AO=OC, OD=OB, and AOD+AOB=180, so cosAOD=-cosAOB.
So, BC^2=OC^2+OD^2+2AO*OBcosAOB (from cosine rule for AOD).
Add the last equation and second equation together:
AB^2+BC^2=AO^2+OB^2+OC^2+OD^2
which is the same as OA^2+OB^2+OC^2+OD^2=AB^2+BC^2.