III.7 Proposition III.7
If on the diameter of a circle a point be taken which is not the centre, and from the point straight lines fall upon the circle: that will be greatest on which the centre is, the remainder of the same diameter will be least, and of the rest the nearer to the diameter through the centre is always greater than the more remote.
Proof
Let AD be a diameter of circle ABCD with centre E, and let
F on AD be distinct from E. From F draw lines FB, FC
to the circumference. Join EB, EC.
In △EBF: EB+EF>FB (I.20). But EB=EA (radii)
and EF+EA=FA, so FA=EF+EB>FB; hence the line FA
along the diameter towards the centre is longer than any other.
The line FD on the other side is similarly the shortest. For
intermediate lines FB vs FC with B closer to A than C,
the SAS inequality I.24 in the radius-line-radius triangles gives
FB>FC when ∠BEF>∠CEF.
lines 74–74 in main.tex
