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Question
In a circle with centre O , chords AB and CD intersets inside the circle at E . Prove that ∠ AOC = ∠ BOD = 2 ∠ AEC.
Solution
Arc AC subtends LAOC at the centre of circle and LABC on the circumference of the cirde .
∴ ∠ AOC = 2 ∠ ABC ...(1)
Similarly, ∠ BOD and ∠ DCB are the angles subtended by the arc DB at the centre and on the circumference of the circle respectively .
∴ ∠ BOD = 2 ∠ DCB ... (2)
Adding ( 1) and (2),
∠ AOC+ ∠ BOD = 2(∠ ABC + ∠ DCB) ... (3)
In triangle ECB ,
∠ AEC = ∠ ECB + ∠ EBC = ∠ DCB + ∠ ABC
From (3),
∠ AOC+ ∠ BOD = 2 ∠ AEC
Hence Proved.
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