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## 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 :

This can be compactly expressed as