Denote by $\omega$ and $\omega^{\prime}$ the incircles of $\triangle A B C$ and $\triangle A^{\prime} B^{\prime} C^{\prime}$ and let $I$ and $I^{\prime}$ be the centres of these circles. Let $N$ and $N^{\prime}$ be the second intersections of $B I$ and $B^{\prime} I^{\prime}$ with $\Omega$, the circumcircle of $A^{\prime} B C C^{\prime} B^{\prime} A$, and let $O$ be the centre of $\Omega$. Note that $O N \perp A C$, $O N^{\prime} \perp A^{\prime} C^{\prime}$ and $O N=O N^{\prime}$ so $N N^{\prime}$ is parallel to the angle bisector $I I^{\prime}$ of $A C$ and $A^{\prime} C^{\prime}$. Thus $I I^{\prime} \| N N^{\prime}$ which is antiparallel to $B B^{\prime}$ with respect to $B I$ and $B^{\prime} I^{\prime}$. Therefore $B, I, I^{\prime}, B^{\prime}$ are concyclic.

Further define $P$ as the intersection of $A C$ and $A^{\prime} C^{\prime}$ and $M$ as the antipode of $N^{\prime}$ in $\Omega$. Consider the circle $\Gamma_{1}$ with centre $N$ and radius $N A=N C$ and the circle $\Gamma_{2}$ with centre $M$ and radius $M A^{\prime}=M C^{\prime}$. Their radical axis passes through $P$ and is perpendicular to $M N \perp N N^{\prime} \| I P$, so $I$ lies on their radical axis. Therefore, since $I$ lies on $\Gamma_{1}$, it must also lie on $\Gamma_{2}$. Thus, if we define $Z$ as the second intersection of $M I$ with $\Omega$, we have that $I$ is the incentre of triangle $Z A^{\prime} C^{\prime}$. (Note that the point $Z$ can also be constructed directly via Poncelet's porism.)

Consider the incircle $\omega_{c}$ with centre $I_{c}$ of triangle $C^{\prime} B^{\prime} Z$. Note that $\angle Z I C^{\prime}=90^{\circ}+ \frac{1}{2} \angle Z A^{\prime} C^{\prime}=90^{\circ}+\frac{1}{2} \angle Z B^{\prime} C^{\prime}=\angle Z I_{c} C^{\prime}$, so $Z, I, I_{c}, C^{\prime}$ are concyclic. Similarly $B^{\prime}, I^{\prime}, I_{c}, C^{\prime}$ are concyclic.

The external centre of dilation from $\omega$ to $\omega_{c}$ is the intersection of $I_{c}$ and $C^{\prime} Z$ ( $D$ in the picture), that is the radical centre of circles $\Omega, C^{\prime} I_{c} I Z$ and $I I^{\prime} I_{c}$. Similarly, the external centre of dilation from $\omega^{\prime}$ to $\omega_{c}$ is the intersection of $I^{\prime} I_{c}$ and $B^{\prime} C^{\prime}$ ( $D^{\prime}$ in the picture), that is the radical centre of circles $\Omega, B^{\prime} I^{\prime} I_{c} C^{\prime}$ and $I I^{\prime} I_{c}$. Therefore the Monge line of $\omega, \omega^{\prime}$ and $\omega_{c}$ is line $D D^{\prime}$, and the radical axis of $\Omega$ and circle $I I^{\prime} I_{c}$ coincide. Hence the external centre $T$ of dilation from $\omega$ to $\omega^{\prime}$ is also on the radical axis of $\Omega$ and circle $I I^{\prime} I_{c}$.

Now since $B, I, I^{\prime}, B^{\prime}$ are concyclic, the intersection $T^{\prime}$ of $B B^{\prime}$ and $I I^{\prime}$ is on the radical axis of $\Omega$ and circle $I I^{\prime} I_{c}$. Thus $T^{\prime}=T$ and $T$ lies on line $B B^{\prime}$. Finally, construct a circle $\Omega_{0}$ tangent to $A^{\prime} B^{\prime}, B^{\prime} C^{\prime}, A B$ on the same side of these lines as $\omega^{\prime}$. The centre of dilation from $\omega^{\prime}$ to $\Omega_{0}$ is $B^{\prime}$, so by Monge's theorem the external centre of dilation from $\Omega_{0}$ to $\omega$ must be on the line $T B B^{\prime}$. However, it is on line $A B$, so it must be $B$ and $B C$ must be tangent to $\Omega_{0}$ as desired.