··

rrxiv is an open protocol. Specification · API · Pulse · AboutMaintained by the rrxiv working group · Privacy-respecting analytics

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

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

Similar triangles are to one another in the duplicate ratio of the corresponding sides.

Let

△ A B C \sim △ D E F

with side ratio

k = B C / E F

. Construct

G

B C

so that

B G = E F \cdot k

(i.e.\ a third proportional, VI.11). By VI.1 the area ratio is

B G : E F = k

along one dimension and the side ratio

k

along the perpendicular, giving total area ratio

k^{2}

in the sense of Definition V.9 (duplicate ratio).