Anya from Tanglin Trust School in Singapore sent in an elegant proof which didn't use vectors (click to enlarge):

Dylan from Brooke Weston and Joshua from Bohunt Sixth Form, both in the UK, used vectors. Joshua set out a good method for approaching the proof. However, Joshua made some assumptions about what the vectors a, b, c and d were, which means Joshua's proof is only valid for those cases. Click here to see Joshua's work.

Dylan proved the statement using vectors. Here is Dylan's work (click to enlarge):