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# geometry

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Four congruent circles are arranged, so that three of them are symmetric and tangent to the fourth circle.  An equilateral triangle is drawn so that each side is tangent to two of the circles.  If the radius of each circle is 1, then find the area of the triangle. Dec 23, 2020

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From the center of  the small  circle on the bottom left draw a radius to the tangent  formed by the side of the equilateral triangle and  draw a segment to the bottom left vertex of the equilateral triangle....call this x

This will form a 30-60-90 right triangle

The vertex angle  of the equilateral triangle will be bisected = 30°

The side opposite of this = radius of a small circle  = 1

So....x  =  twice the radius  =  2

So.....the distance  from the bottom left vertex to the center of the middle  circle  = x + 2 = 2 + 2  = 4

And we can form another triangle  with two sides of 4  and an included angle of 120°

And the side of the equilateral triangle ,S,  wil be  opposite this angle

So.....using the Law of Cosines

S^2  =  4^2  + 4^2  - 2 (16) cos (120°)

S^2  = 32 - 32 (-1/2)

S^2   = 48

S  =  sqrt (48)

So....the area of the equilateral triangle  =

(1/2)  (sqrt (48))^2  sin (60°)  =

(1/2) (48)  sqrt (3)  /2   =

12 sqrt (3)  units ^2   Dec 23, 2020