Inequalities
219. The Triangle's Unbreakable Rule · why two sides always beat the third
In any triangle, the sum of any two sides is always greater than the third side.
Triangle inequality: in any triangle the sum of any two sides is greater than the third side (AB + BC > CA, and likewise for every pair). If this fails for even one pair, the two shorter sides cannot reach each other to close a triangle.
What this lesson covers
Try to break it
Drag A around. As A approaches line BC, AB + AC drops toward BC but never reaches it. The moment AB + AC equals BC, the triangle has flattened to a line — it's no longer a triangle. The triangle inequality (sum of any two sides > the third) is unbreakable.
How you build it
Make a triangle.
- Place point A as the first vertex of the triangle.
- Place point B as the second vertex of the triangle.
- Place point C as the third vertex, not in line with A and B.
- Draw segment AB.
- Draw segment BC.
- Draw segment CA to complete the triangle.
The proof, step by step
Prove that the sum of any two sides of a triangle is greater than the third side.
- Consider any triangle ABC with sides AB, BC, and CA.
- The shortest distance between two points is always a straight line segment.
- Going from B to C via A creates a detour, so the path B → A → C is longer than B → C.
- Thus, AB + AC > BC. By symmetry, AB + BC > AC and AC + BC > AB.
- Therefore, the sum of any two sides of a triangle is always greater than the third side.
Worked example
In a triangle, two sides measure 8 cm and 12 cm. What could be the length of the third side?
By the triangle inequality, the third side must be less than 8 + 12 = 20 cm and greater than 12 - 8 = 4 cm. Only 15 cm satisfies 4 < x < 20.
- 4 cm
- 20 cm
- 15 cm — correct
- 3 cm