1cff-version: 1.2.0
2message: "If you use this rrxiv encoding of Euclid's Elements, please cite as below."
3type: article
4title: "Euclid's Elements, encoded as an rrxiv paper"
5authors:
6 - name: "rrxiv Project"
7date-released: 2026-05-01
8license: CC-BY-4.0
9url: "https://github.com/random-walks/rrxiv-paper-euclid-elements"
10keywords:
11 - Euclid
12 - Elements
13 - geometry
14 - rrxiv
15 - reproducibility
16 - claim graph
17references:
18 - type: book
19 title: "The Thirteen Books of Euclid's Elements"
20 authors:
21 - family-names: "Heath"
22 given-names: "Thomas L."
23 year: 1908
24 publisher: "Cambridge University Press"
25 notes: "Public domain. The translations in this encoding follow Heath with light modernisation."
26
to
AB
extended through
A
to
D
(I.12), so
that the foot
D
falls outside segment
AB
on the far side of
A
.
In the right-angled triangle
BCD
, 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