OK what does Thales Thrm say. If A, B, C are points on a circle, with AB being the diameter then the triangle ABC is a right triangle. So the converse is If we have a right triangle ABC, then AB is the diameter of its circumcircle.
Here is a neat proof using linear algebra.
Refer to the diagram
Now Lets say we have a right triangle ABC, Let AB represent its longest side (Hypotenuse). Now Make a circle with diameter AB with the center of the line AB being the "origin".
Now we need to show that point C also lies on this new circle with diameter AB.
Note that the vector (A-C) is perpendicular to the vector (C-B) . So . Remeber dot product of perpendicular vectors is 0. So we can distribute this through. we get . Now since AB is the diameter and the origin is the center of AB. The Vector A = -B .So substituting which equals since for any vector dot product by itself is equal to its magnitude square , we get or , since ||A|| = radius, this means ||C|| = radius, so C lies on the circle and thus the circle we made with AB as the diameter is indeed the circumcircle for the right triangle ABC. QED.