Proposition III.8 — If a point be taken outside a circle and from the point stra…

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.

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Depends on (4)

III.7Proposition III.7If on the diameter of a circle a point be taken which is not the centre, and from the point straight lines fall upon…
I.20Proposition I.20In any triangle two sides taken together in any manner are greater than the remaining one.
I.24Proposition I.24If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the…
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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