Proposition VI.3 — If an angle of a triangle be bisected and the straight line …

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

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Depends on (4)

I.6Proposition I.6If in a triangle two angles are equal to one another, the sides which subtend the equal angles will also be equal to…
I.29Proposition I.29A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle…
I.31Proposition I.31Through a given point to draw a straight line parallel to a given straight line.
VI.2Proposition VI.2If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle…

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