R/r=N的三角形

费马四平方定理

Fermat's Four Squares Theorem

很容易找到一个面积为6的边长为整数的直角三角形。不存在面积为5的三角形，但存在一个具有有理边的三角形（“\emph{Pythagorean triangle}”）。由于历史的原因，整数如6或5是（平方的部分）的面积的一些毕达哥拉斯三角形被称为“\emph{congruent numbers}”。这些数字实际上很有趣，原因如下：注意序列$\frac14$、$6\frac14$、$12\frac14$。它是一个具有公差6的算术级数，由有理数的平方$(\frac12)^2$、$(\frac52)^2$、$(\frac72)^2$组成。事实上，AP中三个有理平方的公差是一个全等数，每个全等数是算术级数中三个有理平方的公差。由$9^{2}+40^{2}=41^{2}$给出的三角形的面积为$180=5\cdot6^{2}$，如果$x=11{97/144}$，则数字$x-5$、$x$和$x+5$都是有理数平方。回想一下，通过取$x=4uv$，$y=\pm(4u^{2}-v^{2})$，$z=4u^{2}+v^{2}$，可以得到所有具有互质整数边的毕达哥拉斯三角形，其中$u$和$v$是具有$2u$和$v$互质的整数。费马证明，没有AP超过三个平方的有理数。

N-sociable Heronian Triangles

A Heronian triangle is a triangle that has integer side lengths and integer area. Praton and Shalqini [1] define amicable Heronian triangles to be two Heronian triangles where the area of one equals the perimeter of the other, and vice versa; analogous to the concept of amicable numbers. In this paper, we generalize this notion to n-sociable Heronian triangles and characterize them for each n.

A cornucopia of pythagorean triangles

Consider two circles, externally tangential,and with integer radii R1, R2; and with R1>R2.The two circles have three tangent lines in common, one of them being T1T2. If M is the midpoint of T1T2, and K the point of intersection of the lines C1C2 and T1T2;then 16 right triangles are formed(C1 and C2 are the two circle centers), see Figure 1.In Section 6 of this paper, we find the precice form the two integers R1 and R2 must have, in order that the sixteen aforementioned right triangles be Pythagorean.

New relations for the old triangle

In this note we show that in addition to two integers forming a Pythagorean triple, there also exist two irrational numbers in terms of which this Pythagorean triple can also be obtained. We also put forward a relation between these two pairs and the hypotenuse of a Pythagorean triangle.

三角形丢番图问题的一个较简单的解法，其中从顶点开始平分对边的直线可以合理地表示

An easier solution of a Diophantine problem about triangles, in which those lines from the vertices which bisect the opposite sides may be expressed rationally

这是从莱昂哈德·欧拉的拉丁文原文“Solutio facilior problematis Diophantei circa triangulum，in quo rectae ex angulis latera opposita bisecretariat rationaliter exprimantur”的英文翻译。欧拉证明了存在边长为整数的三角形，使得边长与对顶角的平分线的长度为整数，并给出了构造一类这样的三角形的一般方法。