Question

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

Answer 1

consider the following circle.
with center O
let AOB=a°
and ACB=b°
to prove, a=2b

we know that, triangle AOC and BOC are isosceles triangles
(two arms are equal, they are the radii of the same circle)

therefore, from the diagram,
2w +x =180
x=180-2w

and,
2y+z=180
z=180-2y

given that the angle which surround a point, add up to 360°,
a°+x°+z°=360°

therefore,
a +(180-2w) +(180-2y) =360
a -2w -2y=0

a=2(w+y)

but w+y = b

therefore, a=2b
hence proved.

hope this helps.
please correct if there's any mistake

Answer 2

refer the answer given below