Given : AB+AD+AC=AB+BC+BD=BC+CD+AC=BC+AD+BD
To Prove : Quadrilateral ABCD is a rectangle.
Construction : None required.
Proof : We know that AB+AD+AC=AB+BC+BD=BC+CD+AC=CD+AD+BD.
Thus, AB+AD=BC+CD; (1)
AB+BC=CD+AD. (2)
From subracting (2) from (1), we get:
AB-CD=CD-AB.
So,AB=CD.
Similarly,BC=AD.
Thus, ABCD is a parallelogram... (The proof is not completed yet)
Since: AB=BC,
BC is common,
AC=BD,
Angles ABC=BCD.
By proving all the angles equal by the method of congruency above,
All angles must be right angles.
Thus, ABCD is a rectangle. (Proved)
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