With reference to this post,
it was suggested (thanks to user Aretino) that, in the top picture (where the triangle is equilateral and the ellipse pass through two vertex of the triangle and tangent there to two sides), $P$ belongs to the arc of ellipse if $\gamma^2=2\alpha\beta$. Here, $\alpha,\beta,\gamma$ are the segments whose lengths are the distances of $P$ from each side.
My first question is:
How can I prove this?
The second question is related to the second picture:
Does $P'$ belong to the arc of circle if and only if $\gamma'^2=2\alpha'\beta'$?
Notice that, in the first picture the segments $\alpha,\beta,\gamma$ sum up to the altitude of the triangle, whereas, in the second one, the segments $\alpha',\beta',\gamma'$ sum up to the side of the triangle.
The most general issue is:
Is there any relationship between the two scenarios?