Thales' Theorem
Thales of Miletus is often considered the first great Greek mathematician. He seems to be the beneficiary of history's first attribution: "Hey, this is Thales' theorem." See a biography at MacTutor: http://www-history.mcs.st-and.ac.uk/Mathematicians/Thales.html
What he noticed and proved concerned a connection between right triangles and circles. What do you notice?
Why is it true?
Could you prove it's always true? Thales' proof seems to have used that the sum of the angle measures in a triangle are equal to two right angles. (What we call 180 degrees, nowadays.)
More GeoGebra at mathhombre.blogspot.com.