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

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VI.3 Proposition VI.3

If an angle of a triangle be bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle.

Proof

Through

C

draw

C E

parallel to the bisector

A D

(I.31), meeting

B A

produced at

E

. By alternate angles (I.29) and the bisection hypothesis,

∠ A C E = ∠ A E C

, so

A E = A C

(I.6). Applying VI.2 to

△ B C E

with

A D ∥ C E

:

B D : D C = B A : A E = B A : A C

.

lines 74–74 in main.tex

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