Finding a curve which satisfies a special condition about angle We can see that the angle of 
$$\frac{x^2}{a^2}+\frac{y^2}{1-a^2}=1\ \ \ (0\lt a\lt 1)$$
from every point on 
$$C : x^2+y^2=1$$
is $\pi/2$.
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Then, here is my question. 

Question : If the angle of a figure $F$ from every point on $C$ is constant, then can we say that $F$ is an ellipse? 

Here, suppose that $F$ is a simple closed curve on the coordinate plane, and that $F$ is strictly inside of $C$.
For a fixed point $P$ on $C$, let $L_P$ be the set of the lines from $P$ to every point on $F$. Also, let $\theta_P$ be the set of the angles, which include $F$, between every two distinct elements of $L_P$. 
Then, suppose that the angle of $F$ from a fixed point $P$ on $C$ is defined as the maximum element of $\theta_P$.
I've been interested in this question, but I don't have any good idea. Can anyone help?

The points from which a given curve are seen at a fixed angle are called isoptics.  If the angle is a right angle, they are called orthoptics.  These have been investigated by several authors.  The title of a paper by A. Miernowski ("Parallelograms inscribed in a curve having a circle as $\frac \pi 2$-isoptic") in Ann. Univ. Mariae Curie 62 (2008) 105-111, suggests that the answer to your query is negative.
