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

(I.11), with

. Join

,

,

. Through

draw

parallel to

, of length such that

meets the line through

parallel to

at

(I.31). As in II.9,

is a right angle (right-isoceles triangles

and

). Triangle

is also right-angled, with the right angle at

in the configuration where

lies on the extension of

. Apply I.47 twice (to

and to the triangle formed by extending the constructions to

): \[ AD^2 + DB^2 \;=\; 2\cdot AC^2 + 2\cdot CD^2, \] where now

is the half plus the added segment. The derivation parallels II.9 exactly with extension in place of internal cut, and the symmetry is what Heath emphasises.