the three line segments AB, CD and EF are concurrent
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Answer:
Solution
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Given: l∣∣m
Line segment AB,CD and EF are concurrent at P.
Points A,E and C are on line l.
Points D,F and B are on line m.
Refer image,
To prove:
BF
AE
=
BD
AC
=
FD
CE
Proof: In ΔAEP and ΔBFP,
l∣∣m (Given)
∠1=∠2 []Alternate interior angles]
∠3=∠4 [same reason]
∴ΔAEP∼ΔBFP, [By AA similarity criterion]
BF
AE
=
BP
AP
=
FP
EP
(I)
In ΔCEP and ΔDFP,
l∣∣m [Given]
[ALternate interior angles] {
∠7=∠8
∠5=∠6
In ΔCEP and ΔDFP, [By AA similarity criterion]
∴
DF
CE
=
DP
CP
=
FP
EP
(II)
In ΔACP and ΔBDP,
l∣∣m [Given]
[Alternate interior angles] {
∠1=∠2
∠5=∠6
ΔACP and ΔBDP, [By AA similarity criterion]
BD
AC
=
BP
AP
=
DP
CP
(III)
PB
AP
=
BD
AC
=
DP
CP
=
DF
CE
=
FP
EP
=
BF
AE
[From (I),(II) and (III)]]
BD
AC
=
BF
AE
=
DF
CE
Hence proved,
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