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Locus involving chords

Answer Text:
The following properties of chords of a circle are used in construction of loci
(i)Perpendicular bisector of any chord passes through the centre of the circle.
(ii) The perpendicular drawn from a centre of a circle bisects the chord.
(iii) If chords of a circle are equal, they are equidistant from the centre of the circle and vice -versa
(iv) In the figure below, if chord AB intersects chord CD at O, AO = x ,BO = y, CO = m and DO =n then 𝑚 × 𝑛 = 𝑥 × 𝑦