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Source — Euclid's Elements, encoded as an rrxiv paper — rrxiv

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Source — Euclid's Elements, encoded as an rrxiv paper

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III.8 Proposition III.8

If a point be taken outside a circle and from the point straight lines be drawn through to the circle, one of which is through the centre and the others fall on the circle: of the lines falling on the concave circumference, that through the centre is greatest, and the nearer to it always greater than the more remote; and of those falling on the convex circumference, that between the point and the diameter is least, and the nearer to it always less than the more remote.

Proof

The argument mirrors III.7 with the point outside. Let

D

be the external point and

A D

the line through

D

and the centre

E

, meeting the circle at

A

(near) and

C

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III.8 Proposition III.8

Proof

The argument mirrors III.7 with the point outside. Let

D

be the external point and

A D

the line through

D

and the centre

E

, meeting the circle at

A

(near) and

C

(far). For any other line from

D

meeting the circle at

G

(near) and

K

(far), I.20 gives

D G + GE > D E

, and the SAS inequality I.24 again orders the distances by the angles at

E

. The "two lengths per secant" ordering (concave/convex) follows by separating the near and far intersections.

lines 74–74 in main.tex

58 files·489.1 KB·

· 58 files