Question

Prove than the angle subtended by an are of a circle at the centre of the circle is double the angle subtended by it at any point on the remaining part of the circle.

Answer 1

given:ab is arc of circle subtended angle AOB at O.
to prove:Angle AOB =2angle ACB
construction:join AO to extend at B.
proof:in triangle AOC,
exterior angle AOD=angle OAC +angle OCA
=angle OAC =angle OCA
exterior angle AOD=2angle ACO ------(1)
in triangle BBC,exterior angle BOD=angle OCB +angle OBC
angle OBC =angle OCB
angle BOD=2angle OCB-----(2)
add (1)and (2),angle AOD+angleBOD=2angle OCB+2angle ACO
angle AOD +angle BOD =2 (angle OCB + angle ACO)
angle AOB = 2angle ACB
hence proved