The example below seems to suggest No for the inscribed question as well. The line of symmetry is horizontal (dashed). It seems the best aligned isosceles triangle (pink) has area $A_1=\frac{1}{2} (c+\epsilon) b$, while the unaligned, non-isosceles triangle (green) has area $A_2 = \frac{1}{2} c (b+5 \epsilon)$. $A_2 > A_1$ when $5c > b$, which clearly holds.

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          [![InscribedMirror][1]][1]
          ^{Green $\Delta$ area $>$ pink $\Delta$ area.}

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I say "seems" because I have not proved that the pink isosceles triangle is the largest such. (Nor have I proved that the green triangle is the largest inscribed triangle.)

The example below seems to suggest No for the inscribed question as well. The line of symmetry is horizontal (dashed). It seems the best aligned isosceles triangle (pink) has area $A_1=\frac{1}{2} (c+\epsilon) b$, while the unaligned, non-isosceles triangle (green) has area $A_2 = \frac{1}{2} c (b+5 \epsilon)$. $A_2 > A_1$ when $5c > b$, which clearly holds (because $c > b$),

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          [![InscribedMirror][1]][1]
          ^{Green $\Delta$ area $>$ pink $\Delta$ area.}

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I say "seems" because I have not proved that the pink isosceles triangle is the largest such. (Nor have I proved that the green triangle is the largest inscribed triangle.)