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Question

ABCD is a quadrilateral such that AB=AD and CB=CD.Prove that AC is the perpendicular bisector of BD.


Solution

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.
Hence proved.

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