Pythagorean triangles within Pythagorean triangles

TL;DR

This paper explores the conditions under which certain sub-triangles within a Pythagorean triangle are also Pythagorean, providing parametric descriptions and analyzing special cases and their geometric properties.

Contribution

It offers a complete parametric characterization of Pythagorean sub-triangles within a given Pythagorean triangle and analyzes specific cases and conditions for their existence.

Findings

Both sub-triangles BDP and PEA are Pythagorean simultaneously or not at all.

No such point P exists on the hypotenuse of a primitive Pythagorean triangle that makes both BDP and PEA Pythagorean.

Conditions for multiple related triangles to be Pythagorean are explicitly characterized.

Abstract

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.