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I think there is an error in your Pythagoras calculation...
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Doh, that makes it much less elegant :) Thanks for checking and the hint.
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I didn't check further, but this small difference on the perpendicular probably explains the small mismatch at the end with the value given by @didiplaywell.
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The inscribed equilateral triangle has sides of length 6.
The perpendicular has length 5, because Pythagoras 32+52=62.
The extension of the perpendicular must be 1, because 5+1=6.
Platon says the sides of an equilateral triangle's half have ratio 1:2:3.
Therefore, the radius of the inscribed circle is 33+1≈2.732.