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]

Guest Mar 22, 2020

#1**+2 **

**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:

\(\begin{array}{|rcll|} \hline \mathbf{\dfrac{x-1}{1}} &=& \mathbf{\dfrac{1}{3}} \\\\ x-1 &=& \dfrac{1}{3} \\ x &=& 1+ \dfrac{1}{3} \\ \mathbf{x} &=& \mathbf{\dfrac{4}{3}} \\ \hline \end{array}\)

heureka Mar 23, 2020

#1**+2 **

Best Answer

**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:

\(\begin{array}{|rcll|} \hline \mathbf{\dfrac{x-1}{1}} &=& \mathbf{\dfrac{1}{3}} \\\\ x-1 &=& \dfrac{1}{3} \\ x &=& 1+ \dfrac{1}{3} \\ \mathbf{x} &=& \mathbf{\dfrac{4}{3}} \\ \hline \end{array}\)

heureka Mar 23, 2020