It is typically named using one point, which is the center. Sometimes, a circle could be referred to "a circle with diameter AB" in which two points are named. However, you need three non-collinear points to uniquely determine a circle.
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A segment between two points on a circle is
It would be call the ARC.
well its true that every chord contains two points of the circumference of a circle 'cos a chord is the straight line between two points on the circumference
The two points and the centre of the earth define a plane, and the intersection of this plane with the surface of the earth is a circle - the "Great Circle". The shortest distance between the two points is the smaller of the two arcs on this circle.
No. A tangent touches the circle at exactly one point. A line that intersects a circle at exactly two points is a secant.