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If the perpendicular bisector of a chord...

If the perpendicular bisector of a chord AB of a circle PXAQBY intersects the circle at P and Q, prove that arc `PXA cong"arc "PYB`.

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Let AB be a chord of a circle having centre at OPQ be the perpendicular bisector of the chord AB, which intersects at M and it always passes through O.

To prove arc `PXA cong "arc "PYB`
Construction Join AP and BP.
Proof in `Delta APM and DeltaBPM`,
AM=MB [`:'` PM bisects AB]
`anglePMA=anglePMB [:' PMbotAB]`
PM=PM [common side]
`:. Delta APM cong DeltaBPM ` [by SAS congruence rule]
`:.` PA=PB [by CPCT]
`rArr "arc "PXA cong "arc "PYB `
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