A Tale of Three Intersecting Lines
The triangle inequality
Learn the triangle inequality, which decides which three lengths can make a triangle.
Comparing a direct path with the way round is the same as comparing each side with the sum of the other two. If every side is less than the sum of the other two, the lengths satisfy the triangle inequality. 3, 4, 5 does; 10, 15, 30 doesn't.
Look at 10, 15, 30 again: two checks passed and only one failed (30 > 10 + 15). That's always the pattern. Put the lengths in order. Each smaller length is automatically less than the sum of the other two, because that sum includes the longest length. So the only check that can fail is: the longest side against the sum of the two smaller ones.
In short: the two smaller sides together must beat the biggest side. For 4, 5, 8: is 8 < 4 + 5 = 9? Yes. Done; no need to check the rest.
Does passing the check guarantee a triangle? Yes. Draw the longest side as the base and the arcs for the other two. If the two smaller lengths together are more than the base, the arcs reach past each other and must cross, and that crossing is the third corner.
So:
| Lengths | Longest < sum of other two? | Triangle? |
|---|---|---|
| 1, 100, 100 | 100 < 101 ✓ | yes |
| 3, 6, 9 | 9 < 9? ✗ (equal: it lies flat) | no |
| 1, 1, 5 | 5 < 2 ✗ | no |
| 5, 10, 12 | 12 < 15 ✓ | yes |
A handy extra: if two sides are 4 and 7, the third side must be more than 7 − 4 = 3 and less than 7 + 4 = 11.
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