Prove that that a triangle which has one of the angle as `30^0` cannot have all vertices with integral coordinates.

Explanation

Let triangle `ABC` is triangle with one angle =`30^{circ}`

All vertices are not Integral coordinates

Let x, y, a, b& z

slope of line `A B=frac{y}{x}=m_{1}`

slope of line `BC=frac{b}{a}=m_{2}`

Angle between lines `AB times BC=tan 30^{circ}=frac{m_{1}-m_{2}}{1+m_{1} ~m_{2}}`

i.e,` tan 30^{circ}=frac{frac{y}{x}-frac{b}{a}}{1+frac{y}{x} frac{b}{a}}=frac{1}{sqrt{3}} Rightarrow frac{a y-b x}{a x+b y}=frac{1}{sqrt{3}} `
Here LHS is rational RHS is irrational we cannot find any integers a, b, `x, y `that So, all vertices are not Integral coordinates.