﻿ 整数边三角形个数的组合与几何证明方法 Combinatorial and Geometric Proofs of the Number of Triangles with Integer Sides

Vol.04 No.03(2015), Article ID:15892,16 pages
10.12677/AAM.2015.43031

Combinatorial and Geometric Proofs of the Number of Triangles with Integer Sides

Yajing Cai1, Zhengli Tan1, Fugang Chao1, Han Ren1,2

1Mathematics Department, East China Normal University, Shanghai

2Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai

Email: chaofugang@126.com

Received: Jul. 28th, 2015; accepted: Aug. 12th, 2015; published: Aug. 18th, 2015

ABSTRACT

Integer partitions refer to a representation of the positive integer n as a sum of integers. We do not consider the order of terms of the sum. The problem of counting non-congruent triangles with integer sides is just a case of partition of integers. Now, there have been many results about the study of triangles with integer sides problem. In this article, we will solve the problem in two ways. Firstly, we take the common version using the theory of integer partitions to give a proof. Here, we will require generating functions. By using Ferrers diagram, the integer triangles problem will cross to the solution with integers of, while the sum of is equal to the solution of triangles with integer sides problem using the method of generating function. Secondly, we give a geometric approach using triangular coordinates which is easier to understand. Since, we can view as a point in the space, in the triangle cutting off by the planes, ,. Then, the sum of the integral values of corresponds to the number of non-congruent triangles with integer sides. Also, we bring out several further properties, including the number of non-congruent triangles types, such as Isosceles triangles and Equilateral triangles. At the end, we study more about right triangles, acute triangles and obtuse triangles in the non-congruent triangles. But we can just get some relevant properties and conjectures now.

Keywords:Partition of Integers, Ferrers Diagram, Geometric Approach, Triangle Types

1华东师范大学数学系，上海

2上海市核心数学与实践重点实验室，上海

Email: chaofugang@126.com

1. 引言

(1) 问题提出

。这里，表示实数的取整函数。

(2) 背景介绍

(3) 本文概述

2. 主要结果

2.1. 组合方法证明

，则上述条件转化为

Figure 1. Partition of 18

Figure 2. 3-partition of n

(1)

(2)

2.2. 空间格点方法证明

(3)

Figure 3. Function

Figure 4. Lattice point graph

，则记它如上所述对应的网格状三角形为。那么其上有个网格点，即有个非负整数坐标点。内部等边三角形(缩小一圈)为，含有个点。

(2) 当时，代入上面相关公式，则；当时，代入上面相关公式，则，所以

(2) 当时，，所以；当时，，所以；所以

q = 4, N(q) = 3

Figure 5. Lattice point graph with q = 4

q = 5, N(q) = 4

Figure 6. Lattice point graph with q = 6

(1) 整数边长且周长为的不全等三角形中，含等腰三角形、等边三角形的个数为多少？

(2) 能否得出直角三角形、锐角三角形、钝角三角形的相关性质以及个数表达式？

2.3. 等腰、等边三角形个数表达式

Figure 7. 6-partition of triangle

(1) 在直线上，直角三角形的虚线长度为，所含网格点的个数为

(2) 在直线上，直角三角形的虚线长度为，所含网格点的个数为

Figure 8. Lattice point equilateral triangle

(1) 在直线上，直角三角形的虚线长度为，所含网格点的个数为

(2) 在直线上，直角三角形的虚线长度为，所含网格点的个数为

2.4. 直角、锐角、钝角三角形关系

n = 12, n − 3 = 9

Figure 9. Lattice point equilateral triangle

Combinatorial and Geometric Proofs of the Number of Triangles with Integer Sides[J]. 应用数学进展, 2015, 04(03): 246-261. http://dx.doi.org/10.12677/AAM.2015.43031

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(1) 当

(2) 当时，

(1) 当为奇数时，不妨设

(2) 当为偶数时，不妨设