# What you should know about the triangle inequality theorem

The **triangle inequality theorem** states that it is necessary for at least one side of a triangle to be shorter as compared to that of the sum of the other two sides. Why does the **triangle inequality theorem** state so? Well, for the simple reason that if the third side was longer than the other two, it just wouldn’t be possible for the other two to meet!

As per the simplest **triangle inequality theorem** definition, it is necessary for one side of a triangle to be shorter than the sum of the other two sides combined. So basically, here’s what the theorem has to say:

## A look into triangle inequality theorem proof

In order to understand the triangle inequality theorem proof, it is necessary for you to initially understand the shortest distance theorem, which states that the shortest distance from a point that is termed as P to a line that is termed S is the line that is perpendicular to S, whereas it also passes through the point P. Consider the example given below, in which you would notice that the shortest distance is basically the segment that can be termed PR. All other segments are longer than this one.

Now, let’s provide proof for the triangle inequality theorem. For this purpose, you would basically need to draw a triangle ABC, in which the line that happens to be perpendicular to BC passes through the vertex A as well.

Now, you basically need to prove that BA + AC > BC. Here you would note that the shortest distance from the vertex B to that of AE is BE. This proves that BA > BE. This way, it would also be safe to deduce that AC > CE. So, if we combine all of this together, we would have:

BA > BE and

AC > CE

Moving on, you basically need to add the right side and left side of the inequalities, which would give you:

BA + AC > BE + CE

This leads us to believe that BE + CE = BC. This additionally means that BA + AC > BC, which proves the triangle inequality theorem right!

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