Rational distances in triangle

Question

Given triangle with sides of length $3, 4$ and $5$ prove that if $P$ is a point inside the triangle, then rationality of sum of distances from the point $P$ to the vertices implies rationality of sum of distances from the point $P$ to the sides of the triangle. Any hints?

Answer 1

Attempt to solve:

Let $A(3;0); B(0;4); C(0;0); P(x;y)$.

Let $S_1=PA+PB+PC=\sqrt{(x-3)^2+y^2}+\sqrt{x^2+(y-4)^2}+\sqrt{x^2+y^2} \in Q$

Let $d(P;l) - $ distance from a point $P(x_0;y_0)$ to a line $l:ax+by+c=0$. Then $$d(P;l)=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$$ $$AB: \frac x3+ \frac y4=1$$ $$AB:4x+3y=12$$ $$d(P;AB)=\frac{|4x+3y-12|}{5}$$

$$S_2=x+y+\frac{|4x+3y-12|}{5} \in Q?$$