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help with geometry

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There are two circles, one with center A and radius 1, the other with center B and radius 2. The distance AB is 6. A third circle of unknown radius is tangent to both of these circles and there exists a straight line which

(1) is tangent to all three circles, and
(2) intersects the segment AB.

Find the radius of the third circle.

There are two solutions, R and r. Evaluate R - r.

Jul 5, 2020

#1
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There are two solutions, R and r. Evaluate R - r.

Hello Guest!

The distance $$\overline{AB}$$  intersected by the tangent is divided into a and b.

$$a:(6-a)=r_A:r_B\\ a:(6-a)=1:2\\ 2a=6-a\\ a=2\\ b=4$$

$$r_A:(r_A+R)=a:\overline{AB}\\ 1:(1+R)=2:6\\ 6=2+2R$$

$$R=2$$

The radius of the third circle is 2.

There are not two solutions R.

!

Jul 5, 2020
#2
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Hi, asinus! As a matter of fact, there are two solutions; here they are:    ( see diagram )

1)    In both cases, the red line is tangent to all 3 circles.

2)    Both circles, C and D are the tangents to circles A and B.

3)    The red line intersects line AB.

R = 6.75            r = 3.375

Dragan  Jul 6, 2020
edited by Dragan  Jul 6, 2020
edited by Dragan  Jul 6, 2020