Question

o is any point inside a triangle abc.the bisectors of angle aob,angle boc and angle coa meet the sides ab,bc and ca in point d.e and f respectively.show that
1.)ad*be*cf=db*ec*fa

Answer 1

In ∆ AOB, OD is the bisector of angle AOB

OA/OB =AD/DB---------------eq(1)
 
Theorem used here

[The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle]

In ∆BOC .OE is the bisector of angle BOC

OB/OC = BE/EC---------eq(2)

In  ∆COA, OF is the bisector of angle COA

OC/OA =CF/FA-----------eq(3)

Multiplying eq 1, 2, 3 

(OA/OB) * (OB/OC)  * (OC/OA) = (AD/DB) * (BE/EC) * (CF/FA)

1= (AD/DB) * (BE/EC) * (CF/FA)

DB*EC*FA = AD*BE*CF
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AD*BE*CF = DB*EC*FA
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Hope this will help you.....

Answer 2

In ∆ AOB, OD is the bisector of angle AOB

OA/OB =AD/DB---------------eq(1)

Theorem used here

[The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle]

In ∆BOC .OE is the bisector of angle BOC

OB/OC = BE/EC---------eq(2)

In  ∆COA, OF is the bisector of angle COA

OC/OA =CF/FA-----------eq(3)

Multiplying eq 1, 2, 3

(OA/OB) * (OB/OC)  * (OC/OA) = (AD/DB) * (BE/EC) * (CF/FA)

1= (AD/DB) * (BE/EC) * (CF/FA)

DB*EC*FA = AD*BE*CF

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AD*BE*CF = DB*EC*FA

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Hope this will help you.....

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