There was an interesting puzzle by Presh Talwalker in 'MindYourDecisions' about finding the radius of a circle that was cotangent to two larger circles.


I extended the problem by considering adding a second small circle adjacent to the circle of radius 2, (now 16 in my diagram) and wondered what the radius of the two large circles would have to be to make the two small circles have equal radii. The attached diagram will make this more clear. enter image description here I was surprised to find that the ratio of the two circles must be equal to 2.61803398875 which is, of course, the Golden Ratio + 1 The straight lines tangent to the larger circles subtend an angle of 53.130102 degrees. The series of circles can be extended to the right indefinitely and will give pairs of circles of equal radii. See second image. enter image description here Has this been found before? I have looked on the Internet without finding anything.

  • 1
    $\begingroup$ This is very interesting! I strongly recommend you also post this on the Mathematics Stack Exchange (Math.SE). Anything new discovered that contains a special constant such as the golden ratio $\varphi$ should probably be checked out by math experts, which although there are plenty on this site, there are (much) more on Math.SE. $(+1)$ $\color{darkorange}{\bigstar}$ :D $\endgroup$ – Mr Pie Oct 13 '18 at 13:26
  • 3
    $\begingroup$ Thanks, I did post there and someone gave an alternative proof which was far more elegant than mine. math.stackexchange.com/questions/2953796/golden-ratio-plus-1 I guess that many problems with a common proportionality factor will involve the Golden Ratio. Whether this case has been seen before is interesting. $\endgroup$ – George Carey Oct 13 '18 at 15:13

Browse other questions tagged or ask your own question.