Circles on a Line

  1. Consider three identical equilateral triangles, such that one triangle shares a side with each of the other two. A circle is drawn inside each triangle, where all three circles are identical and as large as possible, while keeping the centers of the three circles collinear (all on the same line.) If the height of the triangles is 1, what are the radii of the circles? How far along the side of one triangle is the point of tangency with its internal circle?
  2. Consider three identical squares, where one shares a side with each of the other two (in a right-angle configuration.) As above, identical circles are drawn inside each square such that their centers are collinear and the circles are as large as possible. If the side of the squares is 1, what are the radii of the circles? How far along the side of one square is the point of tangency with its internal circle?
  3. In a 4x4 grid of squares, identical circles are placed in the outer 12 squares, such that they are as large as possible and their centers are all on the same large circle (with center at the middle of the grid.) If the side of the squares is 1, what are the radii of the circles? How far along the side of one square is the point of tangency with its internal circle?
  4. Consider three identical hexagons, where each shares a side with each of the other two. As above, identical circles are drawn inside each hexagon such that their centers are collinear and the circles are as large as possible. If the height of the hexagons is 1 (between opposite sides), what are the radii of the circles? How far along the side of one hexagon is the point of tangency with its internal circle?
  5. Consider three identical circles, where each is tangent to each of the other two. As above, identical circles are drawn inside each circle such that their centers are collinear and the internal circles are as large as possible. If the radii of a large circle is 1, what are the radii of the smaller circles?

Source: Original. The last was suggested by Phillippe Fondanaiche.

Solutions were received from Joseph DeVincentis, Samantha Levin, Colin Bown, Sandra Vollrath, Bill Hewson, Phillippe Fondanaiche, and Stephane Higuere. The answers are:
  1. r = 1/4, t = sqrt(3)/4 = 0.433
  2. r = 1/4, t = 1/4
  3. r = 0.3, t = 0.3, and the radius of the large circle through the centres is (13 * sqrt (2))/10 = 1.838
  4. r = 1/8 = 0.125, t = sqrt(3)/24 = 0.07217
  5. r = 1 - sqrt(3)/2
Colin Bown's general formula which applies to 1, 2, and 4: 
A  C   E
\  /\  /
 \/  \/  
 B   D 

let theta be the size of angles ABC,BCD and CDE.
and L be the length of each segment.

Then the maximal circles with colinear centers have 
radius = L/2 cos(theta/2) sin(theta/2) and are tangent at a point
L/2 cos^2(theta/2)  

thus radius is maximized for the 90 degree arangement
For part 3, from Joseph DeVincentis:
By symmetry, you only need to consider one quadrant of this figure, with three circles in three of the squares on a circular arc about the original center (now the far corner of the 4th square). In the optimum case, all three circles are positioned equivalently (tangent to two sides of the square), but now those circles do NOT have collinear centers.
Let x be the radius of the circles, and (0,0) be the center of the overall figure where the big circle is centered. The first and third circles' centers are located at coordinates (2-x,1-x). The third is located at (1+x,1+x).
So, we need (2-x)^2 + (1-x)^2 = 2 * (1+x)^2, or
4 - 4x + x^2 + 1 - 2x + x^2 = 2 + 4x + 2x^2
5 - 6x = 2 + 4x
3 = 10x

So x = 0.3, or, the circles have radius 3/10 and their points of tangency are 3/10 of a unit from the nearest corners of the squares.

For part 5, from Colin Bown:
let ABC be the centers of the large circle and let abc be the centers of the small circles (a inside A, etc.) s.t. line abc is the line in question (as opposed to bac or acb). Now we see that ab = 1; Aa = 1-r and Bb = 1-r. Angle BAC is 60 degrees so:
tan 60 = 2(1-r)/1 ==> r = 1-sqrt(3)/2

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