Show these three circles share their external common tangent lines

Question

it is a silly problem but I would like to see some ways to show that the external tangent lines from $c_1$ and $c_3$ are also the external tangent lines from $c_1$ and $c_2$:

$c_1$ centered at $A$ passing through $B$.

$BB'$ is a diameter of $c_1$.

$T$ a random point in segment $BB'$.

$c_2$ centered at $B'$ passing through $T$.

$c_3$ centered at $B$ passing through $T$.

To show that $c_1,c_2,c_3$ are all tangent to two lines whose bissector is the support line of $AB$ (who meet in $HI \cap AB$ in the image above).

Answer 1

• We need a following boxed lemma. Let $C$ be a center of homothety which takes one circle to second.

Because of triangle similarity we have $${CO'\over CO} ={R\over r}\implies \boxed{CO = OO'{r\over R-r}}$$

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• Back to the problem.

Let $C_1$ be a center oh homothety which takes $c_2$ to $c_1$ and let $C_2$ be a center oh homothety which takes $c_2$ to $c_3$. All you need to prove is $C_1=C_2$ using boxed formula.