I think you've made an error here and it should be
Which, once you substitute for "a", is identical to what you get from Heron's formula, so doesn't lead anywhere.
By equating two expressions for b from the diagram I arrived at the equation
Then by substituting using you can arrive at a quartic equation in t...it's fairly ugly but like all the other equations, Wolframalpha can solve it and gives leading to .
In case any one is interested..
Of course, knowing the answer, we could factorise the quartic to arrive at our solution but that's not very satisfying.
I did it with vectors. It was probably even more longwinded than BabyMaths effort so I'm not going to bother to post any working.
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