证明三角形任意两条边之和大于第三条边

几何学是数学的一个分支，它研究不同类型的形状、图形和大小。几何学分支处理所见图形中的不同角度、变换和相似性。

三角形

三角形是与三个角、三个边和三个顶点相关联的封闭二维形状。与三个顶点相关联的三角形表示 A、B 和 C，表示为 △ABC。它也可以称为三边多边形或三角形。三角形的一些常见示例是招牌和三明治。

证明三角形任意两条边之和大于第三条边

To prove: The sum of two sides of a triangle is greater than the third side, BA + AC > BC

Assume: Let us assume ABC to be a triangle.

Proof:

Extend the line segment BA to D,

Such that, AD = AC

⇒ ∠ ADC = ∠ ACD

Observing by the diagram, we obtain, 

∠ DCB > ∠ ACD

⇒ ∠ DCB > ∠ ADC

⇒ BD > AB (Since the sides opposite to the larger angle is larger and the sides opposite to smaller angle is smaller)

⇒ BA + AD > BC

⇒ BA + AC > BC.

Hence proved. 

Note: Similarly it can be also proved that, BA + BC > AC or AC + BC > BA

Hence, The sum of two sides of a triangle is greater than the third side.

示例问题

问题 1. 证明上述性质对于最低正整数值成立。

解决方案：

Let us assume ABC to be a triangle. 

Each of the sides is 1 unit. 

Now, 

It is an equilateral triangle where all the sides are 1 each. 

Taking sum of two sides, 

AB + BC ,

1 + 1 > BC

1+1 > 1 

2 > 1

问题 2. 说明直角三角形的这个性质

解决方案：

Let us assume the sides of the right angles triangle to be 5,12 and 13.

Now, 

Taking the smaller two sides, we obtain, 

5 + 12 > 13

17 > 13

Hence, the property holds. 

问题 3. 这个性质是否适用于等腰三角形？

解决方案：

Let us assume a triangle with sides 2x, 2x, and x.

Now, 

Taking the sum of equal two sides, we obtain, 

2x + 2x = 4x 

which is greater than the third side, equivalent to x.