How do you prove that AMON is cyclic?

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Answer 1 · 2017-11-11T05:51:25

given

Construction
$A and O; B and O; C and O; M and O; N and O;$ $Aand O;Band O;Cand O;Mand O;Nand O;$ are joined

In $DeltaONA and Delta ONC$

$OA=OC$ radius of same circle

$ON$ common side

$AN=CN$ given

Hence $DeltaONA ~=Delta ONC$

So $/_ANO =/_CNO=90^@$

similarly $DeltaOMA ~=Delta OMB$

This means $/_AMO =/_BMO=90^@$

So $/_AMO+ANO=90^@$ but these are opposite angles of the quadrilateral $AMON$ .

Hence the quadrilateral $AMON$ is cyclic.