ABCD is a cyclic quadrilateral such that AB=CD , Proove that the diagnols AC and BD are equal.

Given: A cyclic quadilateral ABCD with AB = CD.

To prove: AC = BD.

Proof:

We know that angles in the same segment are equal,

∴∠BDC = ∠BAC   (i)

∠ACD = ∠ABD    (ii)

Consider ΔABP and ΔDCP, we have

∠ABP = ∠DCP    (using (ii))

AB = CD         (given)

∠PAB = ∠PDC     (using (i))

By ASA congruency rule, we have

ΔABP ΔDCP

⇒AP = PD  (CPCT)

⇒PC = BP  (CPCT)

Adding the above two equations,

AP + PC = PD + BP

AC = BD

Hence proved.