, Proposition I.47 gives
\[
BC^2 \;=\; BD^2 + CD^2.
\]
By the binomial-square identity II.4 applied to
BD
cut at
A
(with
BD=BA+AD
as a straight line, since
D
lies on
AB
extended through
A
):
\[
BD^2 \;=\; BA^2 + AD^2 + 2\cdot(BA \cdot AD).
\]
Substitute, and use I.47 in the right-angled triangle
ACD
to
write
AC2=AD2+CD2
; then
AD2+CD2=AC2
, and
substitution gives:
\[
BC^2 \;=\; BA^2 + AC^2 + 2\cdot(BA \cdot AD),
\]
which is the law of cosines as Euclid states it: the square on the
side subtending the obtuse angle exceeds the sum of the squares on
the sides containing it by twice the rectangle on