[High School]How to find out the other two equations?

Question

I have been trying to solve this question

The question asks to prove that- $α=β=γ=30$. I am trying to solve this question by discovering equations. Since,we need to find out the values of three unknown variables.So,we need at least three different equations in order to successfully get the value of those three unknown variables( $α,β$ and $γ$).

I have found out the first equation, $α+β+γ=90..........(i)$

Please help me find out the other two equations.

I will be thankful for help!

NOTE:I am a high school student and English is my second language.

Answer 1

K is a Brocard point and we have the associated relation $\cot A+\cot B + \cot C = \cot \omega$ where $\omega$ is the Brocard angle.

Here this yields $\cot A+\cot B + \cot C=\sqrt 3$.

We further have that in a triangle $\cot A+\cot B + \cot C \le \sqrt 3$ with equality when $\triangle ABC$ is equilateral.

The conclusion now follows