Question

Prove that the angle subtended by an arc is double the angle subtended by it at any point on the remaining part of circle

Answer 1

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Answer 2

Given:
      Arc AB. Point C on the circle is outside AB.

To prove:
 ∠AOB = 2 × ∠ACB

Construction:
Draw a line CO extended till point D.

Proof:
 In ΔOAC in each of these figures,
∠AOD = ∠OAC + ∠OCA (Exterior angle of a triangle is equal to sum of two opposite interior angles)
OA = OC (Radii of same circle)
Thus, ∠ OAC = ∠ OCA (Angles opposite equal sides of a triangle are equal)
∠AOD = ∠OAC + ∠OCA
⇒∠AOD = 2 × ∠OCA
Similarly, in ΔOBC, ∠BOD = 2 × ∠OCB
∠AOD = 2 × ∠OCA and ∠BOD = 2 × ∠OCB
⇒ ∠AOD + ∠BOD = 2 ∠OCA + 2 ∠OCB
∠AOD + ∠BOD = 2 × (∠OCA + ∠OCB)
or ∠AOB = 2 × ∠ACB
Hence, the theorem is proved.