# Twin Incircles

Geometry Level 4

Let $$\triangle ABC$$ be a triangle with sides $$\overline{AB} = 3$$, $$\overline{BC} = 4$$ and $$\overline{AC} = 5$$. Extend the line $$\overline{BC}$$ up to a point $$D$$ so that $$D$$ is closer to $$C$$ than to $$B$$.

This new point $$D$$ is a point such that the radius of the incircle of the triangle $$\triangle ACD$$ is equal to the radius of the incircle of the triangle $$\triangle ABC$$. If the ratio $$\frac{\overline{AD}}{\overline{CD}}$$ can be expressed as $$\frac {x} {y}$$, $$x$$ and $$y$$ being coprime, then find the value of $$x - y$$

×