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 AD 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
DG+GE>DE, 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
