While the author was searching the web, he came across an article by Michael Keyton of IMSA (Illinois Mathematics and Science Academy) called "Theorems of mystery". The phrase is Keyton's own, and he defines such a theorem as "a result that has considerable structure with minimal hypotheses." The simplest of his 10 examples is one that many readers will be familiar with, since it can be investigated at an elementary level and is particularly suited to a dynamic-geometry environment. As Keyton remarks, this theorem "takes an unordered object and creates a very structured one." In this first of a two-part series, the author offers some problems and challenges associated with "The mystery theorem". (Contains 7 figures, 1 table and 1 note.)