Three lines moving at same velocity form a triangle. What is the formula for their lengths at time t?

Question

Three infinite lines on a plane forming a triangle where they cross. The lines are travelling perpendicular to their length at a constant speed $v=1$ such as the triangle is getting smaller.

The lengths of the sides of the triangle at time $t=0$ are $(x_0,y_0,z_0)$. Find the lengths of the sides of $t$.

This is a question I invented while thinking about physics. Extra marks for an intuitive solution without using vectors.

Considering the space-time diagram at some time $t$ all three lines will meet a point forming a triangular pyramid in space-time. Maybe this gives a clue to solving it.

Answer 1

Put the incentre at the origin, the result is just homothety $r(t)=r(0)-t$, for $t<r(0)=r$ (where $r(t)$ is the inradius at time $t$) and hence also on the side lengths $x(t)=x(0)(1-\frac{t}{r(0)})$, etc.

Answer 2

A little more definitive answer..........

$$x_t = x\ (1 - \frac{2 V\cdot t}{(x+z-y)\tan(\frac{A}{2})})$$

$$\text{where}\ z\ge y\ge x\ \text{and}\ (x+z-y)\tan(\frac{A}{2})\ge V\cdot t$$

Substitute $y$ and $y_t$, or $z$ and $z_t$ for $x$ and $x_t$ to get $y_t$ or $z_t$ in the above formula.