A Olympiad geometry question - where two angle bisector are given , two interior angles are given . Find the angles of the triangle .

Question

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 .

For people who require the image

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 .

@ParamanandSingh - there you go . I have uploaded the image . — Math geek, Mar 05 '22 at 12:09

score 0 · Answer 1 · answered Mar 05 '22 at 14:57

0

This is a trigonometric solution.I want help on how to solve it geometrically

answered Mar 05 '22 at 14:57

Tato Kvavadze

1

As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Mar 05 '22 at 14:59
I'm new and do not know how to write here – Tato Kvavadze Mar 05 '22 at 15:17
Check out the link provided in the comments section . They have the geometrical proof . – Math geek Mar 05 '22 at 18:23

A Olympiad geometry question - where two angle bisector are given , two interior angles are given . Find the angles of the triangle .

1 Answers1