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Math: I don't understand wavy lines. Why? What theorem is used?
The vertical diameter theorem is that the perpendicular bisects the chord with the diameter of the chord and bisects the two arcs opposite to the chord.

Inference 1: bisect the diameter (not the diameter) of the chord perpendicular to this chord and bisect the two arcs opposite to this chord.

Inference 2: The perpendicular line of a chord passes through the center of the circle and bisects the arc opposite to this chord.

Inference 3: The diameter of the arc bisecting a chord bisects this chord vertically and bisects another arc opposite this chord.

Inference 4: In the same circle or equal circle, the arcs sandwiched by two parallel chords are equal.