Prove that any four vertices of regular pentagon are concyclic .

Explanation

Let ABCDE be a regular pentagon .
`therefore` AB=BC=CD=DE=EA.
To prove : Any four let A , B ,C ,E points are concyclic.
Construction : Let us join A, C and B , E .
Proof : In triangles ABC ,BC =AE, AB is common to both and `angleABC=angleBAE" " [because` angles of a regular pentagon are equal. ]
`thereforeDeltaABCcongDeltaABE` [ by the S-A-S condition of conguency ]
`thereforeangleACB=angleAEB` , i.e the angles produced at two points C and E on the same side of AB are equal .
`therefore` A , B , C , E are concyclic . [by theorem 39]
Similarly by selecting any other poits , it can be proved that the selected four points are concyclic.
Hence any four vertices of a regular pentagon are concyclic.