Answer : As given , let ABCD be a quadrilateral wherein AB=AD and CB=CD.
To prove : AC is the perpendicular bisector of BD
Consider triangle ADB , as AB=AD, it is an isosceles triangle.
=> by property of isosceles triangle , angle ADB = angle ABD
therefore, triangle ADB is similar to triangle ABD.
Now of similar triangle => side OD = side OB.
=> AB/OB = AD/OD
=> AO is a bisector of BD.
similarly, in triangle BCD, BC= CD
=> it is also isosceles triangle , therefore angle CDB=angle CBD.
and hence , triangle CDB is similar to triangle CBD.
=> side OB= side OD.
=> CD/OD = CB/OB
=> CO is a bisector of BD.
As OA and OC is the bisector of triangle ABD and triangle BCD respectively. Therfore AC is a bisector of BD and
perpendicular to BD.