Geometric Bayesian Update

Today, I present to you Bayes theorem like you have never seen it before.
Take a moment to think about the equation in the Bayes theorem. How would you calculate it using only basic geometry?
Or, to state it more precisely: you are given the unit segment, as well as line segments of lengths equal to P(H), P(E | H) and P(E | ~H) (or the ratio of the last two, if you prefer). How do you get P(H | E) only by drawing straight lines on paper? Can you think of a way that would be possible to implement using a simple mechanical instrument?
I noticed a very neat way to solve this, which is best shown on a diagram:

Have fun with this GeoGebra worksheet.
Your math homework is to find a proof that this is indeed correct (solution).
As an answer to a comment on LessWrong, I also made a pictograph-only version of the diagram:

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