  In this work, we investigate the following question. Given a Pythagorean
triangle BCA, with the right angle at C, let P be a point on the hupotenuse BA;
and let D and E be the perpendicular projections of the point P onto the sides
BC and CA respectively.When is either of the right triangles BDP and PEA
Pythagorean? As it turns out, according to Theorem1, they are either both
Pythagorean, or neither of them is.When they are both Pythagorean,a complete
parametric description of these two triangles is given; in terms of the
parameters that describe the Pythagorean triangle BCA. Later in the paper, we
offer a complete analysis of three special cases:the case wherein the point P
is the midpoint M of the hypotenuse BA; the case when P is the foot I of the 90
degree angle bisector;and the case in ehich the point P is the foot F of the
perpendicular drawn from the vertex C to the hypotenuse. After that, some other
cases are investigated as well.Specifically, we consider the case in which not
only the triangles BDP and PEA are Pythagorean;but the four congruent right
triangles DCP,DEP,DCE,and CEP,are Pythagorean as well.Precise conditions in
order for this to occur, are given in Theorem7. Also,in Theorem 2, part(i), we
show that if the triangle BCA is a primitive Pythagorean triangle, there exists
no point P along the hypotenuse BA, for which both right triangles BDP and PEA
are Pythagorean.
