Source — Euclid's Elements, encoded as an rrxiv paper

to

extended through

to

(I.12), so that the foot

falls outside segment

on the far side of

. In the right-angled triangle

, Proposition I.47 gives \[ BC^2 \;=\; BD^2 + CD^2. \] By the binomial-square identity II.4 applied to

cut at

(with

as a straight line, since

lies on

extended through

): \[ BD^2 \;=\; BA^2 + AD^2 + 2\cdot(BA \cdot AD). \] Substitute, and use I.47 in the right-angled triangle

to write

; then

, 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

(the side on which the perpendicular falls) and

(the segment cut off outside).