Three spheres of radii a, b, and c are tangent to the same plane at points A, B, and C respectively. Each sphere is externally tangent to the other two.
If a < b < c, then which internal angle of triangle ABC is largest and what is its value in terms of a, b, and c?

(In reply to solution by Charlie)

Computed according to the formula I gave, the following table of angle A assumes a to be 1 and gives columns for b up to 4, after which a solution is impossible.  The rows give different values of c--there is no limit to how big c can be if b is sufficiently close to a.  Positions on the table for which b would be greater than c are blank. Positions which represent impossibly large combinations of b and c show hyphens:

   c\b  1.00   1.25   1.50   1.75   2.00   2.25   2.50   2.75   3.00   3.25   3.50   3.75   4.00
 1.00  60.00                                                                                    
 1.25  63.43  67.98                                                                             
 1.50  65.91  71.37  75.52                                                                      
 1.75  67.79  74.06  78.88  82.82                                                               
 2.00  69.30  76.28  81.70  86.17  90.00                                                        
 2.25  70.53  78.17  84.14  89.10  93.38  97.18                                                 
 2.50  71.57  79.82  86.30  91.71  96.42 100.63 104.48                                          
 2.75  72.45  81.27  88.24  94.08  99.20 103.81 108.05 112.02                                   
 3.00  73.22  82.58  90.00  96.26 101.78 106.78 111.42 115.80 120.00                            
 3.25  73.90  83.77  91.62  98.29 104.19 109.58 114.62 119.43 124.09 128.68                     
 3.50  74.50  84.86  93.13 100.18 106.47 112.25 117.70 122.95 128.11 133.29 138.59              
 3.75  75.04  85.86  94.53 101.97 108.63 114.81 120.68 126.41 132.13 137.98 144.18 151.04       
 4.00  75.52  86.80  95.86 103.66 110.70 117.28 123.60 129.84 136.19 142.88 150.30 159.39 180.00
 4.25  75.96  87.67  97.11 105.28 112.70 119.68 126.47 133.28 140.37 148.13 157.47 173.79 ------
 4.50  76.37  88.49  98.30 106.83 114.62 122.03 129.31 136.76 144.74 154.01 167.53 ------ ------
 4.75  76.74  89.27  99.43 108.32 116.49 124.33 132.15 140.31 149.42 161.17 ------ ------ ------
 5.00  77.08  90.00 100.52 109.76 118.32 126.60 135.00 144.00 154.65 172.87 ------ ------ ------
 5.25  77.40  90.70 101.56 111.15 120.10 128.86 137.89 147.90 160.89 ------ ------ ------ ------
 5.50  77.69  91.37 102.57 112.51 121.85 131.10 140.83 152.11 169.98 ------ ------ ------ ------
 5.75  77.96  92.00 103.54 113.82 123.57 133.34 143.88 156.85 ------ ------ ------ ------ ------
 6.00  78.22  92.62 104.48 115.11 125.26 135.60 147.05 162.55 ------ ------ ------ ------ ------
 6.25  78.46  93.20 105.39 116.37 126.94 137.87 150.42 170.91 ------ ------ ------ ------ ------
 6.50  78.69  93.77 106.27 117.60 128.61 140.19 154.06 ------ ------ ------ ------ ------ ------
 6.75  78.90  94.32 107.14 118.81 130.27 142.55 158.14 ------ ------ ------ ------ ------ ------
 7.00  79.11  94.85 107.98 120.00 131.92 144.98 162.98 ------ ------ ------ ------ ------ ------
 7.25  79.30  95.36 108.80 121.17 133.58 147.50 169.61 ------ ------ ------ ------ ------ ------
 7.50  79.48  95.86 109.60 122.33 135.24 150.14 ------ ------ ------ ------ ------ ------ ------
 7.75  79.65  96.34 110.38 123.47 136.91 152.94 ------ ------ ------ ------ ------ ------ ------
 8.00  79.82  96.81 111.15 124.61 138.59 155.97 ------ ------ ------ ------ ------ ------ ------
 8.25  79.98  97.27 111.91 125.73 140.29 159.34 ------ ------ ------ ------ ------ ------ ------
 8.50  80.13  97.71 112.65 126.84 142.02 163.24 ------ ------ ------ ------ ------ ------ ------
 8.75  80.27  98.15 113.38 127.95 143.78 168.23 ------ ------ ------ ------ ------ ------ ------
 9.00  80.41  98.57 114.09 129.05 145.58 180.00 ------ ------ ------ ------ ------ ------ ------
 9.25  80.54  98.99 114.80 130.14 147.44 ------ ------ ------ ------ ------ ------ ------ ------
 9.50  80.66  99.39 115.50 131.23 149.35 ------ ------ ------ ------ ------ ------ ------ ------
 9.75  80.79  99.79 116.18 132.32 151.34 ------ ------ ------ ------ ------ ------ ------ ------
10.00  80.90 100.18 116.86 133.41 153.43 ------ ------ ------ ------ ------ ------ ------ ------
10.25  81.02 100.56 117.53 134.50 155.65 ------ ------ ------ ------ ------ ------ ------ ------
10.50  81.12 100.94 118.19 135.60 158.04 ------ ------ ------ ------ ------ ------ ------ ------
10.75  81.23 101.31 118.85 136.69 160.65 ------ ------ ------ ------ ------ ------ ------ ------
11.00  81.33 101.67 119.50 137.79 163.62 ------ ------ ------ ------ ------ ------ ------ ------
11.25  81.43 102.02 120.14 138.90 167.17 ------ ------ ------ ------ ------ ------ ------ ------
11.50  81.52 102.37 120.77 140.02 172.07 ------ ------ ------ ------ ------ ------ ------ ------
11.75  81.61 102.72 121.40 141.15 ------ ------ ------ ------ ------ ------ ------ ------ ------
12.00  81.70 103.06 122.03 142.28 ------ ------ ------ ------ ------ ------ ------ ------ ------
12.25  81.79 103.39 122.65 143.43 ------ ------ ------ ------ ------ ------ ------ ------ ------
12.50  81.87 103.72 123.26 144.60 ------ ------ ------ ------ ------ ------ ------ ------ ------
12.75  81.95 104.04 123.87 145.79 ------ ------ ------ ------ ------ ------ ------ ------ ------
13.00  82.03 104.36 124.48 147.00 ------ ------ ------ ------ ------ ------ ------ ------ ------
13.25  82.10 104.68 125.09 148.23 ------ ------ ------ ------ ------ ------ ------ ------ ------
13.50  82.18 104.99 125.69 149.49 ------ ------ ------ ------ ------ ------ ------ ------ ------
13.75  82.25 105.30 126.28 150.78 ------ ------ ------ ------ ------ ------ ------ ------ ------
14.00  82.32 105.60 126.88 152.11 ------ ------ ------ ------ ------ ------ ------ ------ ------
14.25  82.39 105.90 127.47 153.49 ------ ------ ------ ------ ------ ------ ------ ------ ------
14.50  82.45 106.20 128.06 154.92 ------ ------ ------ ------ ------ ------ ------ ------ ------
14.75  82.52 106.49 128.65 156.42 ------ ------ ------ ------ ------ ------ ------ ------ ------
15.00  82.58 106.78 129.23 157.99 ------ ------ ------ ------ ------ ------ ------ ------ ------
15.25  82.64 107.07 129.82 159.66 ------ ------ ------ ------ ------ ------ ------ ------ ------
15.50  82.70 107.35 130.40 161.45 ------ ------ ------ ------ ------ ------ ------ ------ ------
15.75  82.76 107.63 130.98 163.40 ------ ------ ------ ------ ------ ------ ------ ------ ------
16.00  82.82 107.91 131.56 165.59 ------ ------ ------ ------ ------ ------ ------ ------ ------
16.25  82.87 108.18 132.14 168.14 ------ ------ ------ ------ ------ ------ ------ ------ ------
16.50  82.93 108.45 132.72 171.36 ------ ------ ------ ------ ------ ------ ------ ------ ------
16.75  82.98 108.72 133.30 176.91 ------ ------ ------ ------ ------ ------ ------ ------ ------
17.00  83.03 108.99 133.88 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
17.25  83.09 109.25 134.46 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
17.50  83.14 109.52 135.03 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
17.75  83.18 109.78 135.61 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
18.00  83.23 110.03 136.19 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
18.25  83.28 110.29 136.78 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
18.50  83.32 110.54 137.36 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
18.75  83.37 110.79 137.94 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
19.00  83.41 111.04 138.53 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
19.25  83.46 111.29 139.11 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
19.50  83.50 111.54 139.70 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
19.75  83.54 111.78 140.30 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
20.00  83.58 112.02 140.89 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
20.25  83.62 112.26 141.49 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
20.50  83.66 112.50 142.09 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
20.75  83.70 112.74 142.69 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
21.00  83.74 112.98 143.30 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
21.25  83.77 113.21 143.91 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
21.50  83.81 113.44 144.53 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
21.75  83.85 113.68 145.15 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
22.00  83.88 113.91 145.78 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
22.25  83.92 114.13 146.41 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
22.50  83.95 114.36 147.05 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
22.75  83.98 114.59 147.70 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
23.00  84.02 114.81 148.35 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
23.25  84.05 115.03 149.01 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
23.50  84.08 115.26 149.68 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
23.75  84.11 115.48 150.36 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
24.00  84.14 115.70 151.04 ------ ------ ------ ------ ------ ------ ------ ------ ------ ------

Posted by Charlie on 2005-09-15 13:59:23