Question 2
Let $ABC$ be a triangle such that
$$\left(\cot\frac A2\right)^2+\left(2\cot\frac B2\right)^2+\left(3\cot\frac C2\right)^2=\left(\frac{6s}{7r}\right)^2,$$
where $s$ is its semiperimeter and $r$ its inradius. Prove that $ABC$ is similar to a triangle whose side lengths are positive integers with greatest common divisor $1$, and determine those integers.
$$\left(\cot\frac A2\right)^2+\left(2\cot\frac B2\right)^2+\left(3\cot\frac C2\right)^2=\left(\frac{6s}{7r}\right)^2,$$
where $s$ is its semiperimeter and $r$ its inradius. Prove that $ABC$ is similar to a triangle whose side lengths are positive integers with greatest common divisor $1$, and determine those integers.