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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 .

Geometry

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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 }

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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