Two unit circles are externally tangent at B. Let AB and BC be diameters of the two circles. A tangent is drawn from A to the circle with diameter BC, and a tangent is drawn from C to the circle with diameter AB, so that the two tangent lines are parallel. Find the distance between the two lines of tangency.
Asymptote Image:[asy]
unitsize(1.5 cm);
pair P, Q, R, T, U;
P = (-2,0);
Q = (0,0);
R = -P;
T = (2/3,2*sqrt(2)/3);
U = -T;
draw(Circle((-1,0),1));
draw(Circle((1,0),1));
draw(P--(T + 0.2*(T - P)));
draw(R--(U + 0.2*(U - R)));
dot("$A$", P, W);
dot("$B$", Q, E);
dot("$C$", R, E);
[/asy]
+26382 Two unit circles are externally tangent at B. Let AB and BC be diameters of the two circles.
A tangent is drawn from A to the circle with diameter BC, and a tangent is drawn from C to the circle with diameter AB,
so that the two tangent lines are parallel.
Find the distance between the two lines of tangency.
I assume:
\(\begin{array}{|rcll|} \hline \mathbf{\dfrac{x-1}{1}} &=& \mathbf{\dfrac{1}{3}} \\\\ x-1 &=& \dfrac{1}{3} \\ x &=& 1+ \dfrac{1}{3} \\ \mathbf{x} &=& \mathbf{\dfrac{4}{3}} \\ \hline \end{array}\)
+26382 Two unit circles are externally tangent at B. Let AB and BC be diameters of the two circles.
A tangent is drawn from A to the circle with diameter BC, and a tangent is drawn from C to the circle with diameter AB,
so that the two tangent lines are parallel.
Find the distance between the two lines of tangency.
I assume:
\(\begin{array}{|rcll|} \hline \mathbf{\dfrac{x-1}{1}} &=& \mathbf{\dfrac{1}{3}} \\\\ x-1 &=& \dfrac{1}{3} \\ x &=& 1+ \dfrac{1}{3} \\ \mathbf{x} &=& \mathbf{\dfrac{4}{3}} \\ \hline \end{array}\)
