Mutually Tangent Circles
Date: 05/22/2000 at 18:26:31 From: Brian Subject: Mutually tangent circles and tangent lines This math puzzle has stumped me. Please give me some answers or at least some suggestions. The puzzle was given with a figure, so I know my figure is accurate. Given: Circle A with radius a Circle B with radius b Circle C with radius c Line S All three circles are tangent to each other No circle is on the interior of another circle Line S is tangent to circles A, B, and C at points D, E, and F, respectively D-F-E What is c (the radius of C) as a function of a and b? The figure below is not drawn to scale (and the actual circles are round). XXXXX XXX XXX XX XXX X X X X X X XX X X X X X X X oooo X X ooo oo X X oo o X X o oo X X oo ooX B X o oX X oo oX X o oXX X o A oXX X o o X X o o X X o o X XX o oo...X X o o. .X XX oo o.. C ..XX X oo oo . . XX XX ooooooooo .... XXXXXXXXXXX _____________D_________F___________E____________________ Thank you for your help.
Date: 05/23/2000 at 06:01:05 From: Doctor Floor Subject: Re: Mutually tangent circles and tangent lines Hi, Brian, Thanks for writing. Let P be the point of intersection of the line through C parallel to DE with the segment AD. Note that AP = a-c and AC = a+c. By Pythagoras' theorem we find that CP = 2sqrt(ac), and thus DF = 2sqrt(ac). In the same way we find EF = 2sqrt(bc) and DE = 2sqrt(ab). Since DE = DF + EF we can derive: 2sqrt(ab) = 2sqrt(ac) + 2sqrt(bc) sqrt(ab) = sqrt(ac) + sqrt(bc) sqrt(ab) = sqrt(c) [ sqrt(a) + sqrt(b)] sqrt(ab) sqrt(c) = --------------- [Take squares] sqrt(a)+sqrt(b) ab c = ----------------- a + b + 2sqrt(ab) If you need more help, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/
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