Question

AB and BC are two chords of a circle intersecting each other at P such that AP=CP show that AB=CD

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Answer 1

Given : AB and CD are chords of a circle and

AP = CP

Construction : Join AC, BC, BD and AD

Now in ∆ APC

AP = CP ....... (1)

⇒∠ PAC = ∠ PCA (∵ Angles opposite to equal sides) ...... (2)

Also we know that angles in the same

segment are equal

⇒ ∠ ACD = ∠ ABD ....... (3)

and ∠ CAB = ∠ CDB ........ (4)

from (2), (3) and (4), we get

⇒ ∠ ABD = ∠ CDB

In ∆ BPD

⇒ BP = DP (∵ sides opposite to equal angles) ........ (5)

Adding (1) and (5), we get

⇒ AP + BP = CP + DP

Hence AB = CD

Answer 2

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Refer to attachment..