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
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Answered by
172
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
-----------------------------------------------------------------------------------------------------
AD*BE*CF = DB*EC*FA
---------------------------------------------------------------------------------------------------
Hope this will help you.....
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
-----------------------------------------------------------------------------------------------------
AD*BE*CF = DB*EC*FA
---------------------------------------------------------------------------------------------------
Hope this will help you.....
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Answered by
13
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
-----------------------------------------------------------------------------------------------------
AD*BE*CF = DB*EC*FA
---------------------------------------------------------------------------------------------------
Hope this will help you.....
e
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