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(2) But

EC

AE

DB

AD

(3) Therefore,

DEC

ADE

BDE

ADE '

'

'

'

? 'BDE 'DEC

(4) But 'BDE and 'DEC are on the

same side of the common base DE. So

they lie between a pair of parallel lines.

HenceBC and DE are parallel.

Theorem 3

The internal bisector of an angle of a triangle divides its opposite side in the

ratio of the sides constituting to the angle.

Proposition : In 'ABCthe line segment AD bisects

the internal angle ‘A and intersects the side BC at D.

It is required to prove that BD : DC =BA :AC.

Construction:Draw the line segment CEparallel to

DA, so that it intersects the side BA produced at E.

Proof:

Steps Justificaltin

(1) Since DAllCE and both BC andAC are their

transversal

and‘AEC ‘BAD

and‘ACE ‘CAD

(2) But ‘BAD ‘CAD

? ‘AEC ‘ACE; ?AC AE

(3) Again, since DAllCE,

AE

BA

DC

BD

?

(4) But AE AC

AC

BA

DC

BD

?

[by construction]

[corresponding angles]

[alternate angles]

[supposition]

[ Theorem 1]

[step (2)]

Theorem 4
If any side of a triangle is divided internally, the line segment from the point of
division to the opposite vertex bisects the angle at the vertex.