From A, the chords are \(AB,AC,AD\). From B, the new chords are \(BC,BD\). From C, the only new chord is \(CD\). This gives six without double-counting.

Opposite equally spaced points are at the ends of diameters, so \(AC\) and \(BD\) pass through the centre.

The four remaining chords join neighbouring equally spaced points. They form a regular quadrilateral, which is a square.

Answer

Six chords. \(AC\) and \(BD\) are diameters; \(AB,BC,CD,DA\) form a square.