I stumbled upon an intriguing geometry puzzle asking whether a hexagon can exist with its vertices on a circle and its internal angles being 70, 90, 110, 130, 150, and 170 degrees in some order. Initially, it seems plausible since the sum of these angles is 720 degrees, which is the requirement for any hexagon. However, the catch lies in the fact that for the vertices to lie on a circle, the hexagon must be a cyclic one, and cyclic polygons have specific properties related to their angles and sides. After some contemplation and a bit of best dissertation writing services UKresearch, I realized that a hexagon with such a set of angles, despite adding up correctly, cannot satisfy the conditions for all vertices to lie on a circle. This puzzle is a clever reminder of how initial assumptions can be misleading in geometry!