How many isosceles, right-angled, INSCRIBED triangles exist in an ellipse?

Consider the following diagram.

Here $D$ is an arbitrary but fixed point on the ellipse, without loss of generality in the first quadrant. $E$ and $G$ are variable points counterclockwise of $D$, and $E'$ and $G'$ are their $90^\circ$ rotations. $DEE'$ and $DGG'$ are then isosceles right-angled triangles. We find the following:

• When the second point is close to $D$, as with $E$, then its rotation ($E'$) lies inside the ellipse.
• When the second point is far from $D$, as with $G$, then its rotation ($G'$) lies outside the ellipse.

So, since motions are continuous, there must be a position for the second point where the third point lies exactly on the ellipse, making an inscribed triangle. Hence there are infinitely many right-angled isosceles triangles inscribed in the ellipse.

As Misha Lavrov says, the exact position for $E$ can be found by rotating the ellipse $90^\circ$ around $D$. $E$ is another ellipse intersection (there may be up to three of them):