In any direction of the sphere, What is the distance from a point to a circle? In any direction of the sphere, What is the distance from a point to a circle?Is there a vertical relationship between the planes of its circle?

There is a point $A$ and a circle $C$ on the sphere. Draw a big circle $D$ from point $A$. There are countless small circles $c$ that pass through point $A$ and tangent to $D$. Some of them will intersect with $C$. The intersection point is $B$. What is the shortest small arc from point $A$ to $B$? Its plane is perpendicular to the $C$ plane, regardless of the direction of $D$?

I guess they are always perpendicular to each other.

On the sphere, my work requires me to find the distance from the outer point of the circle to the circle, and the distance must be in a certain direction.

Conclusion:

Mr. Aretino has proved that when AB is the shortest, AB and C are not vertical. (2) What is the mathematical significance of this conclusion? Are there different shortest directions on the sphere? What is the relationship between this and the shortest distance from a point on the plane to a line in different directions?
@Aretino gave the following interesting answer, but he did not give the shortest AB plane and C plane is vertical, hope he or others can give this conclusion.

Pictorial shortest

Aretino: I thought it was quite evident that the plane of arc $AB$ is not perpendicular to the plane of circle $BE$. In any case I changed again my construction, to show the dihedral angle between those two planes.

Aretino: If circles $C$ and $D$ are perpendicular then of course arc $AB$ lies along circle $D$. You can experiment by yourself with the interactive construction to see if there are other cases. 

Aretino：You didn't answer my question: If $D$ is parallel to $C$, is $AB$ the shortest when $AB$ is perpendicular to $C$?

Mr. @Aretino has proved that when AB is the shortest, AB and C are not vertical.
@Aretino gave the following interesting answer, but he did not give the shortest AB plane and C plane is vertical, hope he or others can give this conclusion.

Pictorial shortest

Aretino: I thought it was quite evident that the plane of arc $AB$ is not perpendicular to the plane of circle $BE$. In any case I changed again my construction, to show the dihedral angle between those two planes.

Aretino: If circles $C$ and $D$ are perpendicular then of course arc $AB$ lies along circle $D$. You can experiment by yourself with the interactive construction to see if there are other cases. 

Aretino：You didn't answer my question: If $D$ is parallel to $C$, is $AB$ the shortest when $AB$ is perpendicular to $C$?

Mr. @Aretino has proved that when AB is the shortest, AB and C are not vertical.