Perhaps these similar triangles are what you're after - simpler and provable without a calculator, though I still used a fair bit of working to prove the side ratios (omitted, tell me if you really want to see it).
There is also a very nice method of dividing the triangle into 3 similar regular trapezoids, but it also gets ugly at the end.
I suspect there might be a further little trick involved to reduce the working - the kind that gets mathematicians all wet around the cosines... Frankly, I still quite like my first sledgehammer approach.
Here's my full working. Involves some trig, but with no evaluation - so it's utterly rigorous. There may be a cheeky way to show it more simply, but this'll do.
Perhaps these similar triangles are what you're after - simpler and provable without a calculator, though I still used a fair bit of working to prove the side ratios (omitted, tell me if you really want to see it).
There is also a very nice method of dividing the triangle into 3 similar regular trapezoids, but it also gets ugly at the end.
I suspect there might be a further little trick involved to reduce the working - the kind that gets mathematicians all wet around the cosines... Frankly, I still quite like my first sledgehammer approach.
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