Understand the problem

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 .    

Source of the problem
Iran Maths olympiad 

Difficulty Level
5 out of 10
Suggested Book

Start with hints

Do you really need a hint? Try it first!

   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/  

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

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 }

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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