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

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VI.2 Proposition VI.2

If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and if the sides of the triangle be cut proportionally, the line joining the points of section will be parallel to the remaining side of the triangle.

Proof

Drop a parallel

D E

from a point

D

A B

to a point

E

A C

, parallel to

B C

. Triangles

△ D B C

and

△ D C E

share the same base

D E

and lie between the same parallels (with

B E

D C

as transversals through I.29, I.37), so they are equal in area. Applying VI.1 to

△ A D E

versus the equal-area companion triangles gives

A D : D B = A E : E C

. The converse runs the argument in reverse via I.39.

lines 74–74 in main.tex

58 files·489.1 KB·

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