Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

PROPOSITION 7.

Let ABCD be a circle, and let AD be a diameter of it;
on AD let a point F be taken which is not the centre of the circle, let E be the centre of the circle, and from F let straight lines FB, FC, FG fall upon the circle ABCD; I say that FA is greatest, FD is least, and of the rest FB is
greater than FC, and FC than FG.

For let BE, CE, GE be joined.

Then, since in any triangle two sides are greater than the remaining one, [I. 20]

But AE is equal to BE;

Again, since BE is equal to CE, and FE is common,
the two sides BE, EF are equal to the two sides CE, EF.

But the angle BEF is also greater than the angle CEF; therefore the base BF is greater than the base CF. [I. 24]

For the same reason

Again, since GF, FE are greater than EG, and EG is equal to ED,

Let EF be subtracted from each;

Therefore FA is greatest, FD is least, and FB is greater than FC, and FC than FG.

I say also that from the point F only two equal straight lines will fall on the circle ABCD, one on each side of the
least FD.

For on the straight line EF, and at the point E on it, let the angle FEH be constructed equal to the angle GEF [I. 23], and let FH be joined.

Then, since GE is equal to EH,
and EF is common,

I say again that another straight line equal to FG will not
fall on the circle from the point F.

For, if possible, let FK so fall.

Then, since FK is equal to FG, and FH to FG,

Therefore another straight line equal to GF will not fall from the point F upon the circle;

Therefore etc. Q. E. D. ^{1 2}