Question

In a circle with centre O , chords AB and CD intersets inside the circle at E . Prove that ∠ AOC = ∠ BOD = 2 ∠ AEC.

Answer 1

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.