勾股三角形内的勾股三角形
Pythagorean triangles within Pythagorean triangles
摘要：
在这项工作中，我们调查以下问题。给定一个直角为C的勾股三角形BCA，设P是最高点BA上的一点，D和E分别是P在边BC和边CA上的垂直投影，那么什么时候直角三角形BDP和PEA是勾股三角形？根据定理1，它们要么都是勾股三角形，要么都不是勾股三角形。当它们都是勾股三角形时，这两个三角形的完整参数描述是根据描述勾股三角形BCA的参数给出的。在本文的后面部分，我们对三种特殊情况作了全面的分析：点P是顶点BA的中点M的情况;点P是90度角平分线的脚I的情况;点P是从顶点C到顶点BA的垂线的脚F的情况。在此之后，我们还研究了其他一些情形，特别是考虑了不仅三角形BDP和PEA是勾股的情形，而且四个全等的直角三角形DCP、DEP、DCE和CEP也是勾股的情形，定理7给出了使这一情形发生的精确条件。在定理2（i）中，我们还证明了：如果三角形BCA是本原勾股三角形，则不存在沿直角三角形BA的点P，使直角三角形BDP和PEA都是勾股三角形。
英文摘要：
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.
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