Proposition: 3.02: Chord Lies Inside its Circle
(Proposition 2 from Book 3 of Euclid's “Elements”)
If two points are taken at random on the circumference of a circle then the straight line joining the points will fall inside the circle.
* Let $ABC$ be a circle, and let two points $A$ and $B$ have been taken at random on its circumference.
* I say that the straight line joining $A$ to $B$ will fall inside the circle.
Modern Formulation
The circle is a convex figure. In particular, if any two points are chosen from the circumference of a circle, and a straight line is constructed on these points, then:
 The segment between the endpoints on the circumference is a chord (i.e. its points are located inside the circle).
 The segment between the endpoints on the circumference is a chord (i.e. its points are located inside the circle).
Table of Contents
Proofs: 1
Mentioned in:
Corollaries: 1
Proofs: 2
Sections: 3
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References
Adapted from CC BYSA 3.0 Sources:
 Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
Adapted from (Public Domain)
 Casey, John: "The First Six Books of the Elements of Euclid"
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"