A triangle is formed by a circle radius and an external point. It is divided by the shown chord. What’s the proportion of the angles α and β?

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## Solution

The proportion α : β is 2 : 1.

Consider the angle 180-β, which is inside the circle as shown. Then choose a point F on the other side, making BDEF a cyclic quadrilateral.

One of the properties of a cyclic quadrilateral is that opposing angles add up to 180°. Therefore, the angle at F is again β.

Finally, use the Inscribed angle theorem to show that α=2β.