Let AB be a rational straight line, let it be cut in extreme and
mean ratio at C, and let AC be the greater segment; I say that each
of the straight lines AC, CB is the irrational straight line called
apotome.

For let BA be produced, and let AD be made half of BA.

Since then the straight line AB has been cut in extreme and mean
ratio, and to the greater segment AC is added AD which is half of
AB, therefore the square on CD is five times the square on DA. [XIII.
1]

Therefore the square on CD has to the square on DA the ratio which
a number has to a number; therefore the square on CD is commensurable
with the square on DA. [X. 6]

But the square on DA is rational, for DA is rational, being half
of AB which is rational; therefore the square on CD is also rational;
[X. Def. 4] therefore CD is also rational.

And, since the square on CD has not to the square on DA the ratio
which a square number has to a square number, therefore CD is incommensurable
in length with DA; [X. 9] therefore CD, DA are rational straight
lines commensurable in square only; therefore AC is an apotome.
[X. 73]

Again, since AB has been cut in extreme and mean ratio, and AC is
the greater segment, therefore the rectangle AB, BC is equal to
the square on AC. [VI. Def. 3, VI. 17]

Therefore the square on the apotome AC, if applied to the rational
straight line AB, produces BC as breadth.

But the square on an apotome, if applied to a rational straight
line, produces as breadth a first apotome; [X. 97] therefore CB
is a first apotome. [p. 451]

And CA was also proved to be an apotome.

Therefore etc.

Q. E. D.