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Three circles of radius #r# units are drawn inside an equilateral triangle of side #a# units such that each circle touches the other two circles and two sides of the triangle. What is the relation between #r# and #a#?

Answer 1

Caroline Aucoin

#r/a=1/(2(sqrt(3)+1)#

We know that

#a = 2x+2r# with #r/x=tan(30^@)#

#x# is the distance between the left bottom vertice and the vertical projection foot of the left bottom circle center.

because if an equilateral triangle's angle has #60^@#, the bisector has #30^@# then

#a = 2r(1/tan(30^@)+1)#

#r/a=1/(2(sqrt(3)+1)#

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Answer 2

Mason Adams

The relation between the radius ( r ) of the circles and the side length ( a ) of the equilateral triangle is:

[ r = \frac{a}{2\sqrt{3}} ]

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