If \(r_1\) and \(r_2\) are positive integers, prove that the coordinates of the points \(P(x,y)\) which divide the line \(P_1P_2\) in the ratio \(r_1:r_2\).

Let \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\) be the coordinates of the points \(P_1\) and \(P_2\) respectively.

The coordinates of the point \(P(x,y)\) which divides the line segment \(P_1P_2\) in the ratio \(r_1:r_2\) can be found using the section formula:

\[x = \frac{r_1x_2 + r_2x_1}{r_1 + r_2}\]
\[y = \frac{r_1y_2 + r_2y_1}{r_1 + r_2}\]

Since \(r_1\) and \(r_2\) are positive integers, the sum \(r_1 + r_2\) is also a positive integer. Therefore, the coordinates of the point \(P(x,y)\) are also integers.

Thus, the coordinates of the points \(P(x,y)\) which divide the line \(P_1P_2\) in the ratio \(r_1:r_2\) are integers.

Let \(P(x,y)\) be the point that divides the line segment \(P_1P_2\) in the ratio \(r_1:r_2\). Let \(P_1(x_1,y_1)\) and \(P_2(x_2,y_2)\). Then, using the section formula, we have:
$$x=\frac{r_1x_2+r_2x_1}{r_1+r_2}$$
$$y=\frac{r_1y_2+r_2y_1}{r_1+r_2}$$

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