Questions on Inscribed Triangles - inspired by Reuleaux

Repeating a question recorded at https://mathoverflow.net/questions/432417/more-on-triangles-inscribed-in-convex-regions-with-one-vertex-fixed. We also continue the previous post here.

Note: Motivation for these questions is the existence of infinitely many constant width closed curves other than the circle. And as was noted in above linked page, one can think of analogous questions on smallest triangles that contain C.

1. Are there convex shapes C other than (any) ellipse such that: the area of the max area inscribed triangle with one vertex at P remains constant as P moves around boundary of C? If "yes", what could be said about them?

Note: Indeed, for any ellipse, as a point P moves around its boundary, the function: "the area of the max area inscribed triangle with one vertex fixed at P" is a constant. For each P, there is an inscribed triangle with centroid coindient with center of ellipse and for which the ellipse is the steiner circumellipse.

2. Are there convex shapes C other than the circle such that: the perimeter of the max perimeter inscribed triangle with one vertex at P remains constant as P moves around boundary of C?