Question

How to find an integer point inside a convex polygon?

Answer 1

Let $A_1,A_2,A_3,A_4......A_n,$ is the point of our polygon .
for closed polygon it is necessary that ,
$A_{n+1}=A$

Now, Let F is the point inside the polygon and T is the point outside the polygon .
If F exists inside the polygon only when ,
∠A₁FAA₂ + ∠A₂FA₃ + ∠A₃FA₄ + ......+ ∠AₙFAₙ₊₁ = 360° { because F forms a circle and we know, centre of circle is 360° }
so, we can say that conditions of any point inside the polygons is
$\sum^n_{k=0}{\angle A_k FA_{k+1}} = 360$

For T point
$\sum^n_{k=0}{\angle A_k TA_{k+1}} \neq 360$