(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
andAEC BAD
andACE 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.