This is one of my favorite problems of all time because it has such a surprising geometric solution. It appeared as problem six at the International Mathematical Olympiad in 2001 and was solved by only 27 out of 473 of the world's smartest high school students.
Links I promised in the video:
Cyclic quadrilaterals: https://en.wikipedia.org/wiki/Cyclic_quadrilateral#Supplementary_angles
Ptolemy's Theorem:
https://www.youtube.com/watch?v=bJOuzqu3MUQ