Descartes' theorem (1964)

If four mutually tangent circles have curvature

*k*_{i}(for*i*= 1,...,4), (ki=1/ri, Where ri is radius)(1) |

(2) |

*two*solutions. Ignoring the degenerate case of a straight line, one solution is positive and the other is either positive or negative; if negative, it represents a circle that circumscribes the first three (as shown in the diagram above).

Other criteria may favor one solution over the other in any given problem.

**Special case**

If one of the three circles is replaced by a straight line, then one

*k*

_{i}, say

*k*

_{3}, is zero and drops out of equation (1). Equation (2) then becomes much simpler:

(3) |

**Example:**

If three circles of radius 1 touches one another then find the radius of the circle which can be inserted within these circles, touching all the three.

**[CAT]****Solution:**

k4=3+2root(3)[positive for inscribed circle] therefore radius=1/[3+2root(3)]=[2-root(3)]/root(3)

PS:One can use conventional method too.

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