Given :  7 points with integer COORDINATES on a 2-D plane,

To Find : what can be the maximum number of equilateral triangles they can form

Solution:

Lets take any 3 points out of  7 points

A ( a , b)  , B = ( c , d) , C = ( e , f)

AREA of Triangle =  (1/2) | a (d - f)  + c (f - b) + e (b - d) |

now  integer coordinates Hence  Area of triangle will be rational number

Area of Equilateral Triangle = (√3 / 4) (side)²

(side)²   = lets take AB   = ( a - c)² + ( b - d)²  which again a rational number

Hence Area of Equilateral Triangle =  (√3 / 4) * rational number

as  rational number /4  = also Rational number

= √3  * Rational number

IRRATIONAL number *  non zero Rational number = irrational number

Hence   Area of Equilateral Triangle =  irrational number

But from integer coordinates on a 2-D plane we get Area of triangle will be rational number

Hence both the STATEMENTS contradicts

So its not possible to have an Equilateral triangle with integer coordinates on a 2-D plane

So There can not be any triangle  irrespective of number of points

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