On Inscribed and Escribed Circles of Right Triangles, Circumscribed Triangles, and the Four-Square, Three-Square Problem

For more than twenty years, I have been studying the fascinating relationships between right triangles and their inscribed and escribed circles. An inscribed circle is one that is tangent to all three sides of a triangle and whose center lies inside the triangle. An escribed circle is one that is tangent to one of the sides of the triangle and to the extensions of the other two sides and whose center lies outside the triangle. Every triangle has one inscribed and three escribed circles, as shown in figure 1.