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Three circles are drawn, so that each circle is externally tangent to the other two circles. Each circle has a radius of 2. A triangle is then constructed, such that each side of the triangle is tangent to two circles, as shown below. Find the perimeter of the triangle.

 Jun 7, 2020
 #1
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Let  A =  lower-left-hand vertex of the triangle.

Let  B =  top vertex of the triangle.

 

Let  P =  center of the lower-left-hand circle.

Let  Q =  center of the top circle.

 

Let  X =  point of tangency of AB with circle(P).

Let  y =  point of tangency of AB with circle(Q).

 

[The left diagonal will be A-X-Y-B.]

 

Since each circle has radius 2:

 

PQ  =  4

PX  =  2

QY  =  2

XY  =  4

 

Triangle(APX) is a 30o - 600 - 90o triangle with AP the hypotenuse.

Since PX =2,  AP = 4,  and  AX  =  2·sqrt(3).

YB  =  AX  =  2·sqrt(3). 

 

AB  =  AX + XY + YB  =  2·sqrt(3) + 4 + 2·sqrt(3)  =  4 + 4·sqrt(3)

 Jun 7, 2020

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