Relations between Various Magnitudes Which Serve as Signs of a Right-angled Triangle

Relations between different magnitudes in a triangle, such as the radius of the circumcircle of the triangle, the
radius of the incircle of the triangle, the radii of the excircles the triangle (which are tangent to one side from the
outside and the continuations of the two other sides), as well as their relations with the side lengths the triangle
and certain segments in the triangle, are a fascinating subject. We present a proven scheme of 15 relations
between the sides of a triangle, the radii of incircles and excircles of the triangle, the area and half the perimeter
of the triangle, which serve as signs of a right-angled triangle, together with the 16th relation which is the
Pythagorean Theorem. The relations are investigated dynamically using the computerized software.