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June 2, 2019

Lattice point inside a triangle

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Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.22.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]Given a triangle ABC with three lattice vertices . it is known that no more lattice point lies on the edges . only one lattice point D is inside the triangle . prove that D is centroid of that triangle .    

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Iran Maths olympiad 
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Start with hints

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   Do you know Pick’s theorem ?  NO!  then read this post first and return to the problem again  https://www.cheenta.com/a-proof-from-my-book/  

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 Join AD , BD , & CD  NOW , can you do it ?

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To prove D is centroid it is suficient to prove that [ ABD] = [ BDC] = [ ADC ]   where [.] denotes the bounded area . but why ? { think yourslfe , use the fact that medians  devide the triagle in six equal part }

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Now applying pick's theorem we get , [ABD]= [ BDC] = [ ADC ] = 0.5

Because in each of these three  triangle , three are 3 lattice vertex and lattice point is inside the triangle . so  area = 0 + 3/2 - 1 = 0.5 

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Connected Program at Cheenta

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Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

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Similar Problem

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