Art Deco

Geometry Level 5

Given above is Δ ( A B C ) \Delta(ABC) with an area of 60. Points A ′ , B ′ A',B' and C ′ C' lies on line segments B C , A C BC, AC and A B AB so that A C ′ = B C ′ AC'=BC' , A ′ C = 2 B A ′ A'C=2BA' and B ′ A = 3 C B ′ B'A=3CB' . A A ′ , B B ′ AA',BB' and C C ′ CC' intersect eachother at points M , N M,N and P P . Find the area of Δ ( M N P ) \Delta(MNP) .


The answer is 6.

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Leah Smith by Leah Smith , 25, USA

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3 solutions

Leah Smith
Dec 5, 2015

This problem is dedicated to Menelaus' Theorem

11 Helpful 0 Interesting 0 Brilliant 0 Confused

Maybe Barycentric Coordinates can also be a good idea for people who are not familiar with Routh or Menelaus.

Lorenc Bushi - 5 years, 6 months ago

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Hm, I think Menelaus' Theorem is more basic than Barycentric Coordinates (or at least, when I learnt them).

Calvin Lin Staff - 5 years, 6 months ago
Jon Haussmann
Dec 6, 2015

An instance of Routh's Theorem .

5 Helpful 0 Interesting 0 Brilliant 0 Confused

@Jon Haussmann you make me feel so embarrassed... I only took advantage of Menelaus' Theorem

Leah Smith - 5 years, 6 months ago

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Rouths theorem has been derived from Menelaus' Theorem

Manisha Garg - 5 years, 6 months ago

Well, you did it from more fundamentals. That is not any thing to be embarrassed about.

Niranjan Khanderia - 5 years, 6 months ago

Nice we did it the same way :)

Jun Arro Estrella - 5 years, 6 months ago

Used Routh's Theorem from WiKi.

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