From fb page of Nelson Tunala

 

https://artofproblemsolving.com/community/c4t48f4h2603744_angles_in_isosceles_wanted_am2_24o_kyiv_city_olympiad_2004_87_

 

https://artofproblemsolving.com/community/c6t48f6h3479955_angle_chasing_in_an_equilateral_triangle

 I do not understand why almost nobody tries to solve synthetically! See attached:

https://www.facebook.com/photo/?fbid=664212039157837&set=pcb.664212079157833, a lemma

 

Geometria Super Top, Manuel Duarte, https://www.facebook.com/photo/?fbid=3893966467415125&set=gm.1853469585423402&idorvanity=830782544358783

 

https://www.facebook.com/photo/?fbid=8883447381723660&set=gm.523343850755063&idorvanity=184162418006543