Let (P,r) denote a circle with center P and radius r.
Circles (A,a) and (B,b) are externally tangent with
a less than b.
Let CD and EF be the common external tangents to
circles (A,a) and (B,b) with C and F on circle (A,a)
and D and E on circle (B,b).
What is the area of trapezoid CDEF in terms of
a and b?
put A:(0,0) and B:(a+b,0)
then we can determine the coordinates for C,D,E, and F as
C: ( a(a-b)/(a+b) , 2a*sqrt(ab)/(a+b) )
F: ( a(a-b)/(a+b) , -2a*sqrt(ab)/(a+b) )
D: ( a*(a+3b)/(a+b) , 2b*sqrt(ab)/(a+b) )
E: ( a*(a+3b)/(a+b) , -2b*sqrt(ab)/(a+b) )
from these you can easily get CD, DF, CF, DE, and EF. and from these using Heron's formula on triangles CDF and DEF you can get the area of CDEF which comes out to
8*a*b*sqrt(ab)/(a+b)
which is much nicer looking that I was expecting it to be
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Posted by Daniel
on 2010-07-28 16:26:33 |
analytical solution
