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

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is acute, the foot

D

falls within the segment

A B

, between

A

and

B

. Apply Proposition II.7 to

A B

cut at

D

:

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

. In the right-angled triangle

B C D

, I.47 gives

B C^{2} = B D^{2} + C D^{2}

, and in

△ A C D

likewise

A C^{2} = A D^{2} + C D^{2}

. Subtracting (Common Notion 3) the first from the third:

A C^{2} - B C^{2} = A D^{2} - B D^{2}

. Combining with II.7 and rearranging (Common Notion 3 + Common Notion 2): \[ AC^2 \;=\; AB^2 + BC^2 - 2\cdot(AB \cdot BD), \] which is the acute-angle form of the law of cosines: the square on the side subtending the acute angle is less than the sum of the squares on the sides containing it by twice the rectangle on

A B

and

B D

(the segment cut off within).

lines 74–74 in main.tex