Question

We have `A(0,0) B(21,0), C(0,21)`.How to find number of points that have both coordinates integer numbers inside triangle `ABC`?

Answer 1

`(x,y)` with `0 le x le 21` and `0 le y le 21-x`, with `x,y \in \mathbb{N}`

Explanation:

The triangle is delimited by the two axes and the line `y=-x+21`.

So, both `x` and `y` range from `0` to `21`, and every point in the triangle lies between the line and the `x` axis.

So, if `x=0`, we have `y = -(0)+21`, and so every point like

`(0,0), (0,1), ..., (0,21)`

belong to the triangle.

if `x=1`, we have `y=-1+21=20`, and so every point like

`(1,0), (1,1), ..., (1,20)`

So, the following points are inside the triangle:

`(x,y)` with `0 le x le 21` and `0 le y le 21-x`, with `x,y \in \mathbb{N}`