Question

Given 7 points with integer coordinates on a 2-D
plane, what can be the maximum number of
equilateral triangles they can form? (equilateral
triangles are the triangle with all sides equal)
Answer: enter your answer here​

Answer 1

Answer:

see down

Step-by-step explanation:

In two dimensions there’s no equilateral triangle with integer coordinates, i.e. whose vertices are lattice points. Let’s prove that.

Suppose ABC is an equilateral triangle whose coordinates are integers. In complex numbers A,B and C are Gaussian integers, complex numbers whose parts are integers.

In an equilateral triangle, if we rotate side AB around vertex A 60∘60∘ counterclockwise or clockwise, we’ll land on AC. We express that:

C−A=(B−A)e±iπ/3C−A=(B−A)e±iπ/3

Since

e±iπ/3=cosπ3±isinπ3=12±i3–√2e±iπ/3=cos⁡π3±isin⁡π3=12±i32

we can’t have both C−AC−A and B−AB−A be Gaussian integers, because there has to be 3–√3 somewhere. That’s our contradiction; we conclude there are no equilateral triangles whose vertices are lattice points in the two D Cartesian plane.

That’s not true in three dimensions: Consider triangle ABC,

A(1,0,0),B(0,1,0),C(0,0,1)A(1,0,0),B(0,1,0),C(0,0,1)

It’s equilateral:

AB=BC=AC=2–√

Answer 2

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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