Faganano's problem

We are looking for the inscribed circle with minimal perimeter.
Reflecting the perimeter of the inscribed triangle creates a path between D and its doubly reflected image D''. This path needs to be a straight line.
Now it only remains to find the best place for D.
Do this by trial and error, and see if you can come up with any ideas as to where the point is geometrically.

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|D D''|

Can you prove your hypothesis?

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