Question

How do you prove that AMON is cyclic?

Answer 1

Construction
`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.