Quadrilateral with given angles

If you are looking for the proof by elementary geometry, please see Hiroshi Saito(斉藤浩)'s great work about generalized Langley's problem ( https://www.gensu.co.jp/saito/challenge/pdf/3circumcenter_d20180609.pdf ). He introduced the amazing skill named "3 circumcenter method" invented by Ms. aerile_re(pen name), and you can find the solution of this problem in the article (Q1).

my geometric solution: http://www.davdata.nl/math/geopuzzle28.html

arc calculation