The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.

Try this
Adjust the triangle by dragging the points A,B or C. Notice how the longest side is always shorter than the sum of the other two.

In the figure above, drag the point C up towards the line AB. As it gets closer you can see that the line AB is always shorter than the sum of AC and BC. It gets close, but never quite makes it until C is actually on the line AB and the figure is no longer a triangle.

The shortest distance between two points is a straight line. The distance from A to B will always be longer if you have to 'detour' via C.

To illustrate this topic, we have picked one side in the figure above, but this property of triangles is always true no matter which side you initially pick. Reshape the triangle above and convince yourself that this is so.

A triangle cannot be constructed from three line segments if any of them is longer than the sum of the other two.

For more on this see
Triangle inequality theorem converse.
- Triangle definition
- Hypotenuse
- Triangle interior angles
- Triangle exterior angles
- Triangle exterior angle theorem
- Pythagorean Theorem
- Proof of the Pythagorean Theorem
- Pythagorean triples
- Triangle circumcircle
- Triangle incircle
- Triangle medians
- Triangle altitudes
- Midsegment of a triangle
- Triangle inequality
- Side / angle relationship

- Perimeter of a triangle
- Area of a triangle
- Heron's formula
- Area of an equilateral triangle
- Area by the "side angle side" method
- Area of a triangle with fixed perimeter

- Right triangle
- Isosceles triangle
- Scalene triangle
- Equilateral triangle
- Equiangular triangle
- Obtuse triangle
- Acute triangle
- 3-4-5 triangle
- 30-60-90 triangle
- 45-45-90 triangle

- Incenter of a triangle
- Circumcenter of a triangle
- Centroid of a triangle
- Orthocenter of a triangle
- Euler line

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