 Problem #117  Posted Thursday, May 10, 2001
 The Kissing Circles (back to top)
 Inside the
(orange) unit circle we
fit two blue circles of radius 1/2.
 a) The yellow circle is tangent
to the two blue circles and the
 inner edge of the orange
circle. What is its radius, a?
 b*) The green circle is tangent
to one of the blue circles, the
 yellow circle, and the orange
circle. What's its radius, b?
 c*) If the orange circle
is centered at (0, 0) what are the coords
 of the centers of the four
inner circles? (The blue ones are
easy.)
 Solution:
Allen Druze was the last entrant
so I'll make him semifamous:

 Let diameter COD run through the centers
of the two blue circles.Since
 CD = 2 ,CO = 1 and OD = 1. The radius
of both blue circles equal 1/2,
coordinates of (1/2,0) and (1/2,0). Draw
a perpendicular line from the
top of circle a to center point 0. Let the radius of circle a
= a and the
distance from circle a to center 0 be x. Then 2a + x = 1 ( note:
by
connecting the centers of the circles I can prove congruency
by s.a.s. hence
center a lies on the line that is perpendicular to COD.). Connect the center
of circle a to the first blue circle(left side) and you will
have a right
triangle that will yield (1/2)^2 + (a+x)^2 = (a + 1/2)^2, since
x = 1 
2a , by substituting for x, the first equation will yield a = 1/3, or
coordinates (0, 2/3) . Draw a line
tangent thru pt C and construct a semi
circle equal to circle a that is tangent to both circle b as
well as one of
the blue circles. Let b equal the radius of the green circle,
therefore
 a + 2b + a = 1 , since a = 1/3 , b
= 1/6, hence circle b has coordinates
(1/2, 2/3).


 Check out my new page
 on this circle problem!
