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 
Topic
Geometry 

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 

Connected Program at Cheenta

I.S.I. & C.M.I. Entrance Program

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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