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#?

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Answer 1 · 2016-11-25T15:40:47

#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)#

so

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

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