In the figure, there are two circles intersecting each other at A and B. Prove that the line joining their centres bisects the common chord.
Let C(O, r) and C'(O', r) be two circles having the common chord AB.
Join OA, OB, O'A and O'B
In ΔAOO' and ΔOBO',
OA = O'A, OB = O'B and OO' = OO'
... ΔOAO' @ DOBO' ... ∠AOP = ∠POB
Now in ΔAOP and ΔBOP,
OA = OB, OP = OP
and ∠AOP = ∠BOP ... ΔAOP ≅ ΔBOP
... AP = BP.
Join OA, OB, O'A and O'B
In ΔAOO' and ΔOBO',
OA = O'A, OB = O'B and OO' = OO'
... ΔOAO' @ DOBO' ... ∠AOP = ∠POB
Now in ΔAOP and ΔBOP,
OA = OB, OP = OP
and ∠AOP = ∠BOP ... ΔAOP ≅ ΔBOP
... AP = BP.
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