Infinite when chords are parallel to the circle's diameter
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No, a circle can't be a parallelogram. A circle is a curve. A parallelogram is a quadrilateral with two pairs of parallel sides constructed with four line segments. The line segments are straight, and the circle is a continuous curve.
Do you mean, like, the Earth? The equator. If you mean the line going from side to side across the middle of a circle then... the diameter of a circle
infinite
Pair of radii
A pair of parallel lines, an ellipse or a circle.