On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions: $\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$ BD$. $\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$. $\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally. $\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$. Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.