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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.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

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Similar Problem

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