Geometric Bayesian Update

There's no reason why P(E|H) and P(E|~H) must sum to 1, but I can't move the lower right corner without the whole diagram rescaling.

Two-word proof: Prin'f gurberz. (Nebhaq gur gevnatyr, plpyvpnyyl, jr unir: rivqrapr, cevbe bqqf, erpvcebpny cbfgrevbe bqqf.)

I think this would be clearer with only the triangle where all the action is happening, and without the stuff on the left whose only job is to put the whole thing into a rectangle. You can still have the prior odds and the evidence on perpendicular axes: make it a right-angled triangle and let what's now the right-hand edge of the rectangle turn into the diagonal.

Not a proof:

Gur boivbhf ohg oehgr sbepr jnl gb qb guvf jbhyq or svaqvat gur rdhngvba bs gur gjb yvarf sebz gur gjb evtug pbearef (obgu rnfl orpnhfr bs fvzvyne gevnatyrf naq/be xabja yratguf/pbbeqvangrf), svaq gur cbvag jurer gurl vagrefrpg, jevgr gur rdhngvba sbe gur yvar cnffvat guebhtu vg naq gur ybjre yrsg pbeare, fbyir sbe gung yvar uvggvat gur evtug yvar, gura fvzcyvsl. V unira'g qbar vg, ohg guvf fubhyq onfvpnyyl jbex qverpgyl.

Vf gurer fbzr oevyyvnag jnl bs fubjvat vg ol zrer fvzvynevgvrf? V qvqa'g frr bar vzzrqvngryl.