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.