Proposition III.25 — Given a segment of a circle, to describe the complete circle…

Proof

Let

A B C

be the given segment with chord

A C

and arc through

B

. Pick

B

on the arc; join

A B

B C

. Bisect

A B

D

and

B C

E

(I.10). At

D

and

E

erect perpendiculars to

A B

and

B C

respectively (I.11). By III.3 / III.9 these perpendiculars both pass through the centre, so their intersection

F

is the centre. With centre

F

and radius

F A

describe the full circle.

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Full neighborhood

Depends on (5)

III.3Proposition III.3If in a circle a straight line through the centre bisect a straight line not through the centre, it also cuts it at…
III.9Proposition III.9If a point be taken within a circle, and more than two equal straight lines fall from the point on the circle, the…
I.10Proposition I.10To bisect a given finite straight line.
I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.
I.15Definition I.15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among…

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