Similar Triangle through the Incenter

FG is constructed parallel to side AC of triangle ABC and through the incenter of ABC.
Try different side lengths AB, BC, AC and look at the perimeter of the triangle BFG.
Can you make any conjecture about a formula for the perimeter?

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BC
AC
AB
BF
BG
FG

Assuming your conjecture is true, what is the scale factor between the two (similar) triangles. (i.e. what is the ratio AC::FG).
Can you prove your conjecture?

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