Problem, aops, https://artofproblemsolving.com/community/c6t48f6h2267253_equal_angles_wanted_trapezoid_equal_segments_right_angle_related

 My solution: 

aops, https://artofproblemsolving.com/community/c4t48f4h3520153_angle_is__always_45

Still using inversion, 

 

Geometria Super Top, Hakan Ersoz, https://www.facebook.com/groups/830782544358783

 A nice an easy to understand synthetic solution to this beautiful problem

FB page of Kenan Rustemov, https://www.facebook.com/photo?fbid=802536598658713&set=pcb.802536625325377

 My solution

aops, https://artofproblemsolving.com/community/c6t48f6h2547832_triangle_with_60ampdeg_angle_projection_of_othocenter_onto_median

My solution

aops, https://artofproblemsolving.com/community/c6t48f6h144817_find_the_angle

 My solution:

Mohammed Bayezid Beg, Beautiful Geometry Problems, https://www.facebook.com/photo?fbid=628826876746088&set=gm.1384088496293486&idorvanity=970715314297475

 My solution

Mohammed Bayezid Beg, Beautiful Geometry Problems, https://www.facebook.com/photo/?fbid=628733423422100&set=gm.1383988929636776&idorvanity=970715314297475

 My solution

aops, https://artofproblemsolving.com/community/c6t48f6h3418669_ltc__if_ltb45o_and_bp__pc_1__2_apc__60o__without_trigonometry

 My solution, almost synthetic, since using sin 30=0.5 cannot be synthetic

Ahmet Cetin, Romantics of Geometry, https://www.facebook.com/groups/parmenides52

 My solution:

FB page of Kenan Rustemov, https://www.facebook.com/photo/?fbid=664212039157837&set=pcb.664212079157833

 My solution:

FB page of Haka Ersoz, https://www.facebook.com/photo/?fbid=455543863586888&set=pcb.455543956920212

 My solution: 

aops, https://artofproblemsolving.com/community/c4t48f4h3514062_pmo26_qualifying_ii15_corner_to_interior

 

Nelson Tunala, Geometria Super Top, https://www.facebook.com/photo/?fbid=9170031573065238&set=gm.1886029895500704&idorvanity=830782544358783

 My solution:

Math. stackexchange, https://math.stackexchange.com/questions/3704315/find-angle-cad-x-for-triangle-abc-d-is-a-point-inside-triangle?rq=1

 

https://www.facebook.com/photo/?fbid=122111791262752641&set=a.122096449862752641, fb page of Well Mae

 

Gogeometry, problem 1586

 See below the synthetic solution (if using a 30 degs angle may be deemed as synthetic).

fb page of Nelson Tunala

 

Adventitious angles, https://artofproblemsolving.com/community/c4t48f4h3494669_adventitious_angles_synthetic

For 'double angle lemma' you may see my book 'Rose Thorns - Geometry Gewels' 

 

My newest book, problems of angle calculus in poligons

 You may find it for purchase at https://ebooks.gil.ro/acasa/203-cactus-bloom-geometry-problems-collection-ebook.html

https://www.facebook.com/photo/?fbid=844548767828287&set=a.408188904797611

 A rather difficult synthetic solution:

https://artofproblemsolving.com/community/c6t48f6h3487462_sad_geometry

 A nice solution (synthetic) below

Aops square

 

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