Circle Theorem Proof: Angle subtended at centre of circle is twice angle at circumference

In this article I'll be showing you how to prove that the angle subtended at the centre of the circle is twice the angle at the circumference. It is recommended that you watch the video below whilst reading to comprehend the proof fully.

Firstly:

2x + A = 180

2y + B = 180

This means that:

A = 180 - 2x

B = 180 - 2y

Let's say that the angle beside 'A' is 'C', and that the angle next to 'B' is 'D'. Because half way around a circle is 180 degrees, we'd have to say that:

A + C = 180

B + D = 180

This implies that C = 180 - A and that D = 180 - B. Since A = 180 - 2x and B = 180 - 2y:

C = 180 - (180 - 2x) = 180 - 180 + 2x = 2x

D = 180 - (180 - 2y) = 180 - 180 + 2y = 2y

As C = 2x and D = 2y:

C + D = 2x + 2y = 2(x+y)

Now, since this is the case, it is true that the angle subtended at the centre of the circle is twice the angle at the circumference.

• Article written by Tiago Hands: https://www.instagram.com/tiago_hands

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