If the chord of contact of tangents from a point on the circle $x^2+y^2=r_1^2$ to the circle $x^2+y^2=r_2^2$ touches the circle $x^2+y^2=r_3^2$, then $r_1, r_2, r_3$ are in:

Let $C_{1 }: x^2+y^2=r_1^2$

      $C_2: x^2+y^2=r_2^2$

       $C_{3 }: x^2+y^2=r_3^2$

Let $AB$ be the chord of contact of tangents from a point $P\left(x_1,y_1\right)$  on the circle $C_1$ to the circle $C_2$.

So the equation of $AB$

           $T=0$

$\Rightarrow x{x}_1+y{y}_1-r_2^2=0$

Since line $AB$ touches the circle $C_3$.

By condition of tangency,

The distance of the line $AB$ from the centre of the circle$C_3=r_3$,

$\frac{|0.x_1+0.y_1-r_2^2|}{\sqrt{x_1^2+y_1^2}}=r_3$

 As $P\left(x_1,y_1\right)$ lies on $C_1$,

$x_1^2+y_1^2=r_1^2$

$\Rightarrow \frac{|-r_2^2|}{\sqrt{{r_1}^2}}=r_3$
$\Rightarrow r_2^2=r_3.r_1$

So, $r_1,r_2, r_3$ are in GP.