G.130 complete a circle from a given segment — Heath

III.25

Given a segment of a circle, to describe the complete circle of which it is a segment.

Let ( A B C ) be the given segment of a circle; thus it is required to describe the complete circle belonging to the segment ( A B C ), that is, of which it is a segment.

For let / A C / be bisected at D, let / D B / be drawn from the point D at right angles to / A C /, and let / A B / be joined;

the angle < A B D ) is then greater than, equal to, or less than the angle < B A D ).

First let it be greater; and on the straight line / B A /, and at the point A on it, let the angle < B A E ) be constructed equal to the angle < A B D ); let / D B / be drawn through to E, and let / E C / be joined.

Then, since the angle < A B E ) is equal to the angle < B A E ), the straight line / E B / is also equal to / E A /. [G.27]

And, since / A D / is equal to / D C /, and / D E / is common, the two sides / A D /, / D E / are equal to the two sides / C D /, / D E / respectively; and the angle < A D E ) is equal to the angle < C D E ), for each is right;

therefore the base / A E / is equal to the base / C E /.

But / A E / was proved equal to / B E /;

therefore / B E / is also equal to / C E /;

therefore the three straight lines / A E /, / E B /, / E C / are equal to one another.

Therefore the circle drawn with centre E and distance one of the straight lines / A E /, / E B /, / E C / will also pass through the remaining points and will have been completed. [G.107]

Therefore, given a segment of a circle, the complete circle has been described.

And it is manifest that the segment ( A B C ) is less than a semicircle, because the centre E happens to be outside it.

Similarly, even if the angle < A B D ) be equal to the angle < B A D ), / A D / being equal to each of the two / B D /, / D C /,

the three straight lines / D A /, / D B /, / D C / will be equal to one another, D will be the centre of the completed circle, and ( A B C ) will clearly be a semicircle.

But, if the angle < A B D ) be less than the angle < B A D ), and if we construct, on the straight line / B A / and at the point A on it, an angle equal to the angle < A B D ), the centre will fall on / D B / within the segment ( A B C ), and the segment ( A B C ) will clearly be greater than a semicircle .

Therefore, given a segment of a circle, the complete circle has been described.

Q. E. F.

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