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

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paper/figures/fig-i-32.pngimage · 8904 bytesRaw

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parallel to

A D

or

B E

(I.31), meeting

D E

at

F

. On

C B

as side construct the square

C B K G

inside

A D E B

(a copy of II.4's construction), with

H K

parallel to

A B

. By II.4 the square

A D E B

on

A B

equals the square on

A C

plus the square on

C B

plus twice the rectangle

A C \cdot C B

. Add the square on

C B

to both sides of this identity (Common Notion 2): \[ AB^2 + CB^2 \;=\; AC^2 + 2\cdot CB^2 + 2\cdot(AC\cdot CB). \] But

2 \cdot C B^{2} + 2 \cdot (A C \cdot C B) = 2 \cdot C B (C B + A C) = 2 \cdot C B \cdot A B

(Definition II.1; II.1). Hence

A B^{2} + C B^{2} = 2 \cdot (A B \cdot C B) + A C^{2}

, as required.