Triangular Inequalities
The sum of two sides of a triangle is greater than the third side. A natural conclusion from it is that any side of a triangle has to be GREATER than the difference of the other two. In other words, let the three sides of a triangle be a, b and c. We have:
a + b > c > | a - b |
This inequality is super helpful for us to find the range of a side in a triangle.
2024 AMC 8 Problem 6: This one is not exactly about triangles, but the idea of the shortest path remains the same.
Triangular inequality often serves to provide some constraints for counting problems.