Construction of Segment on Given Circle Admitting Given Angle

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In the words of Euclid:

From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle.

(The Elements: Book $\text{III}$: Proposition $34$)



Let $ABC$ be the given circle, and let $D$ be the given rectilineal angle.

Let $EF$ be drawn tangent to $ABC$ at $B$.

Let $\angle FBC$ be constructed equal to $\angle D$.

Then $BC$ is a straight line which forms the base of a segment which admits an angle equal to $D$.


Let $A$ be selected anywhere on the circle opposite the segment in question.

From Angles made by Chord with Tangent‎, $\angle BAC = \angle FBC$.

But $\angle FBC = \angle D$.

Hence the result.


Historical Note

This proof is Proposition $34$ of Book $\text{III}$ of Euclid's The Elements.