O is a point in the interior of a square ABCD such that `Delta OAB ` is an equilateral triangle . Show that `Delta OCD ` is an isoceles triangle .

Explanation

Given O is a point in the interior of a square ABCD such that `Delta OAB ` is an equilateral triangle .

Construction join OC and OD
To show `Delta OCD ` is an isosceles trianlge
Proof Since,AOB is an isosceles triangle .
` therefore angle OAB =angle OBA =60 ^(@)`
Also `angle DAB =angle CBA -angle OBA =90^(@)-60 ^(@)`
i.e `angle DAO= angle CBO =30^(@)`
in ` Delta AOD and Delta BOC `
AO=BO [given ]
[all the side of an equilateral triangle are equal ]
`angle DAO =angle CBO` [proved above ]
and AD=BC [side of a square are equal ]
`therefore Delta AOD cong Delta BOC ` [by SAS congruence rule ]
Hence OD=OC [by CPCT ]
OC=OD
Hence `Delta COD` is an isosceles triangle