PROPOSITION 25.
Given a segment of a circle,
to describe the complete circle of which it is a segment.
Let
ABC be the given segment of a circle; thus it is required to describe the complete circle belonging to the segment
ABC, that is, of which it is a segment.
For let
AC be bisected at
D, let
DB be drawn from the point
D at right angles to
AC, and let
AB. be joined;
the angle ABD is then greater than, equal to, or less than the angle BAD.
First let it be greater; and on the straight line
BA, and at the point
A on it, let the angle
BAE be constructed equal to the angle
ABD; let
DB be drawn through to
E, and let
EC be joined.
Then, since the angle
ABE is equal to the angle
BAE,
the straight line EB is also equal to EA. [I. 6]
And, since
AD is equal to
DC, and
DE is common,
the two sides AD, DE are equal to the two sides CD, DE respectively; and the angle
ADE is equal to the angle
CDE, for each is right;
therefore the base AE is equal to the base CE.
But
AE was proved equal to
BE;
therefore BE is also equal to CE; therefore the three straight lines
AE,
EB,
EC are equal to one another.
Therefore the circle drawn with centre
E and distance one of the straight lines
AE,
EB,
EC will also pass through the remaining points and will have been completed. [
III. 9]
Therefore, given a segment of a circle, the complete circle has been described.
And it is manifest that the segment
ABC is less than a semicircle, because the centre
E happens to be outside it.
Similarly, even if the angle
ABD be equal to the angle
BAD,
AD being equal to each of the two
BD,
DC,
the three straight lines DA, DB, DC will be equal to one another,
D will be the centre of the completed circle, and ABC will clearly be a semicircle.
But, if the angle
ABD be less than the angle
BAD, and if we construct, on the straight line
BA and at the point
A on it, an angle equal to the angle
ABD, the centre will fall on
DB within the segment
ABC, and the segment
ABC 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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