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No, not all chords of a circle pass though the center of that circle. Any cord that does pass through the center of the circle is called diameter of that circle.
The only chords that are diameters are the chords that go through the center of the circle. All of the other chords are shorter.
They are equidistant from the center of the circle
Generally, no. All circles contain an infinite number of chords, as a chord can be created between any two points on the circle. With an infinite number of points on the circle we can create an infinite number of chords.
Infinite when chords are parallel to the circle's diameter