A Olympiad geometry question - where two angle bisector are given , two interior angles are given . Find the angles of the triangle . This is a problem from 2017 Bulgaria Selection Test for the Junior Balkan Mathematical Olympiad:

Given a triangle $\triangle ABC$ and $AA_1$, $BB_1$ are angle bisectors. If angle $AA_1B_1$ = $24^{\circ}$, angle $BB_1A_1$ = $18^{\circ}$.Find out the other angles of the triangle .


So - Angle C can be easily deduced with basic geometry . But from here the trouble begins - I tried various constructions (like, trying to get the angles in a parallelogram or something). At last, I got the answer by using sine law repeatedly.
But this wasn’t satisfactory for me as I always work with pure Euclidean way of geometry . By using trigonometry - I many have got the answer but not the required satisfaction .
Hence, I asked my colleagues, tried myself over a Month ( I don’t like to give up ) and here I am. Anybody with a pure geometry ( Euclid geometry ) proof. You have my appreciation .

I want your help in solving this task. I know the trigonometric solution. How to solve it in another way, maybe you can help me in advanceenter image description here
