Three circles touch each other perfectly — how big is the fourth that touches all three?
Draw three circles of any size so that each one touches the other two, then draw a fourth circle nestled in the gap between them, also touching all three. Given only the sizes of the first three circles, is there a way to calculate the exact size of the fourth, without any measuring or trial and error?
Reveal the answer
Yes: define each circle's 'bend' as the inverse of its radius, and the four bends always satisfy a simple quadratic relationship first stated by René Descartes in 1643. In 1936, chemist and Nobel laureate Frederick Soddy rediscovered the formula and published it as a poem in Nature titled 'The Kiss Precise,' since mutually tangent circles are sometimes called 'kissing circles.' The formula has two valid answers, since a small circle can fit snugly in the gap, or a large one can wrap around all three from outside.
— Frederick Soddy, restating René Descartes' 1643 theorem, The Kiss Precise — Nature, 1936