**Problem**

Suppose there are four circles lying in a plane, such that each circle is mutually tangent to the other three circles. Let the radius of each circle be . Prove that

**Hint:** Given three angles where , then .

**Solution**

Let , , , , , and .

Furthermore, let there be three angles , , and .

By applying the cosine law on triangle ADB, we get

Expanding each bracket and rearranging yields

Solving for yields

Using the same method, we can also obtain

Let

Using the trigonometric identity from the hint gives

Expanding and rearranging the above equation yields

Divide both sides of the equation by :

Replacing the variables for in terms of gives

Expanding the left-hand side of the equation yields

Expanding the right-hand side of the equation yields

Subsequently,

Divide both sides by :

Add on both sides:

This can be compactly expressed as

**Additional Remark**

Solving the result for gives two solutions: the radius of the incircle (inner circle), and the radius of the excircle (outer circle).

**Incircle radius**

**Excircle radius**

**Historical Note**

Rene Descartes found the theorem without proof and sent it to Princess Elizabeth of Bohemia in 1643. In Europe, the theorem was first proven by Jakob Steiner nearly 200 years later in 1826. In Edo Japan, this theorem first appeared in a sangaku problem from 1796. The theorem was explained in Hashimoto Masataka's book *Sanpo Tenzan Shogakusho* (筭法點竄初學抄) in 1830.