Triangle and lines

We can choose two statements from three 3 ways:

1) The line goes through the midpoint of the incircle of the triangle and the line divides the triangle's area into two equal areas.

The line divides the sides of the triangle into the $x, y, z, u, w$ parts. From the statements we get:

$\frac{rv}{2}+\frac{rw}{2}+\frac{rx}{2}=\frac{ry}{2}+\frac{rz}{2}.$

I can divide it with $\frac{r}{2}$ , because it isn't $0$ . I get:

$v+w+x=y+z$

2) The line goes through the midpoint of the incircle of the triangle and the line divides the triangle's perimeter into two equal parts.

Now we know that

$v+w+x=y+z$

Now if I multiply it with $\frac{r}{2}$ , I get:

$\frac{rv}{2}+\frac{rw}{2}+\frac{rx}{2}=\frac{ry}{2}+\frac{rz}{2}.$

3) The line divides the triangle's area into two equal areas and the line divides the triangle's perimeter into two equal parts.

The triangle cuts the RX line segment from the line. We know that

$v+w+x=y+z$

Now if I multiply it with $\frac{r}{2}$ , I get:

$\frac{rv}{2}+\frac{rw}{2}+\frac{rx}{2}=\frac{ry}{2}+\frac{rz}{2}.$

So the $XPR$ "broken" line cuts the triangle into two equal parts. Since the line is also cuts the triangle into two equal parts, the line goes through $P$ .