Page 5 - LN

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E
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Given: A triangle ABC and a line intersecting sides AB in D and BC in E

B
d

B

C
C
A

e

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t
B
r
r
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B
i

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n
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such that .
p
B
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o
a
E
D
s
a
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i

l
l
t
l
To prove: DE is parallel to BC..

t
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h
a
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g
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a

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e

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B

.
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Proof: Let us start by assuming that DE is not parallel to BC.

m
m
n
r
b
st
n
u
h
a
F
s
a
C
h
T h e n t t h e r r e m u st b e a n o t t h e r r l l i i n e t t h r r o u g h D t t h a t t i i n t t e r r se ct s A C a t t F s u ch
ch
T
u
Then there must be another line through D that intersects AC at F such
e
e
ct
se
e
m
n
A
h
e
s
u
h
e
o
a
h
h
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e
D
n
n
o

F
D

C
a
a
i
B
p
s
r

.
e
l
o
t

l
that DF is parallel to BC.
l
t

o
r
y
Therefore, by Basic Proportionality Theorem, we have,,
n
e
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o
a
r
e
h
t
f
o
p
o
t
i
T
l
r
e
e
B
o
w

h

e
r
si
e
m

a
,
c
e
,
v

b
P
a
h
r
T
y

But, (given)

Thus,
Or,
Therefore, we get,
T h e r e f o r e , w e g e t ,
FC = EC.
This is possible only if points E and F are coincident.
T h i s i s p o ssi b l e o n l y i f p o i n t s E a n d F a r e co i n ci d e n t .
Hence, DF and DE are one and the same segmentts.
H e n c e , D F a n d D E a r e o n e a n d t h e sa m e se g m e n
H e n c e , D E i s p a r a l l e l t o B C
Hence, DE is parallel to BC..

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