Is the answer 5?

Geometry Level 4

In Δ A B C \Delta ABC , A D , B E , C F AD , BE , CF are concurrent such that D , E , F D,E,F are points on B C , A C , A B BC,AC,AB respectively.

If C E = 1 CE=1 , C D = 2 CD=2 , B D = 3 BD=3 , A F = 4 AF=4 , A E = 6 AE=6 ,then what is the length of B F BF ?

Note: Figure not to scale.


The answer is 1.

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Nihar Mahajan by Nihar Mahajan , 20, India

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2 solutions

Nihar Mahajan
May 14, 2015

Since the cevians are concurrent , by Ceva's theorem ,

B D C D × C E A E × A F B F = 1 3 2 × 1 6 × 4 B F = 1 ⇒ B F = 1 \dfrac{BD}{CD} \times\dfrac{CE}{AE} \times\dfrac{AF}{BF} =1 \\ \dfrac{3}{2} \times\dfrac{1}{6} \times\dfrac{4}{BF} =1 \\ \Rightarrow \boxed{BF=1}

16 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice solution!

Kartik Bhardwaj - 6 years, 1 month ago

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Thanks! :) :)

Nihar Mahajan - 6 years, 1 month ago

Very good solution. Sad I did not know ceva's theorem.....

Debmalya Mitra - 5 years, 11 months ago
Avinash Singh
May 20, 2015

Well... I hope you don't require any background for this. Simple logic can give you the answer. Ratio of one sides length to the other sides ratio will give you the ratio for the third side.

1 Helpful 0 Interesting 0 Brilliant 1 Confused

Didn't understand ur solution Avinash :/

Saibal Chakraborty - 5 years, 9 months ago

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