ABCD is a quadrilateral such that AB = AD and CB = CD. Prove that AC is the perpendicular bisector of BD.
Given: ABCD is quadrilateral
And AB = AD, CB = CD
To prove : AC is bisector at BD
In ABC and ADC
AB = AD (Given)
BC = CD (Given)
AC = AC (Common)
ABC ADC (by SSS congruency)
Then by CPCT,
ÐBAC = ÐDAC …(i)
Now in ABO and ADO
AB = AD (Given)
AO = AC (common)
BAO = DAO (from i)
ABO DAO (by SAS congruency)
Then by CPCT,
AOB = DOA
BOD = 180°
AOB + AOD = 180° (BOD = AOB + AOD)
AOB + AOB = 180° (AOB AOD)
AOB = 90° (AOB = AOD)
Hence, proved.