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Let D,E,F be three points on the sides BC, AC and AB of a triangle ABC such that AD, BE and CF are concurrent if BD*CE*AF = K*DC*EA*FBThen the value of K is explain with reason

Let D,E,F be three points on the sides BC, AC and AB of a triangle ABC such that AD, BE and CF are concurrent if BD*CE*AF = K*DC*EA*FBThen the value of K is explain with reason

Grade:11

1 Answers

Vedant
35 Points
3 years ago
We take ,
CD = BC/a, AE = AC/b, BF/c where a,b and c are ratios
Then BD = BC – CD = BC – BC/a = BC(a – 1)/a
         CE = AC – AE = AC – AC/b = AC(b – 1)/b
         AF = AB – BF = AB – AB/c = AB(c – 1)/c
 
Therefore,
         BD * CE * AF = [AB * BC * AC * (a – 1) * (b – 1) * (c – 1)] / abc           ...1
and   CD * AE * BF = k *  [AB * BC * AC] / abc                                              ...2
 
On compairing 1 and 2, we find that k = (a – 1) * (b – 1) * (c – 1)
where a = BC / CD, b = AC / AE and c = AB / BF

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