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Math Help - 3 circles tangent to another and all 3 inside a 4th.

  1. #1
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    3 circles tangent to another and all 3 inside a 4th.

    [IMG]file:///C:/DOCUME%7E1/JBURTO%7E1/LOCALS%7E1/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]

    The large Circle is Radius 4.51 and the 3 smaller ones 2.06. While not super perfectly to scale….it is very close. On to my question:
    Where can I find equations to prove/disprove that 3 or more circles with radius R1 will fit within the bigger circle radius r2?
    All 3 are perfect circles and are the same size. while i do not Understand Descartes’ Theorem (never learned that in HS) i was thinking it my be what i am after. Excerpt for the net:if four mutually tangent circles have curvature a, b, c, and d, then…
    (a + b + c + d )2 = 2(a2 + b2 + c2 + d 2 ).

    My line of thinking is i have 3 mirrors and wish to place them all in a perfectly circular hoop and then fill in with stained glass.
    What must the 3 little radius be to fit in the 4th? they do not have to touch the hoop and can be much smaller.


    Thanks in advance for your time, and double thanks to any that can help. some links on what i found but was not making headway with:
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  2. #2
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    a second diagram to note what i am asking

    This is my 3rd diagram, showing 3,4,5 circles in 1. i have found using the "Geometry Expressions Demo" that for 3 in 1 circle, the radius is ~46.4% of the large circle. the others are 4in1: Y2=41.3%, and 5in1 Y4=37%

    Ignore large circle is a Radius 4 Doc, they were poor attempts at trying to convey my question.

    Question restated: for x#of circles inside 1, what is the equation for x circles radii ? for example, R1(.5-(.04(x-2))) where R1 is the outer circle radius, and x is the # of circles past 2
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    Last edited by Ratheden; July 17th 2010 at 05:30 PM. Reason: Edited for clarity and Rewrite of my question/idea
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