When Two Sticks Can't Reach

Where it usually goes wrong

"The sum of two sides must be greater than the third" is checked as three additions and a comparison, with no sense of what going wrong actually looks like — so a student can correctly compute that 2 + 3 is not greater than 6 and still not know why that means no triangle exists, because nothing has shown them two sides physically failing to meet.

Try this

• Drag C to the right, stretching the base. Watch the blue arc (everywhere exactly a from B) and the warm arc (everywhere exactly b from C) drift apart as the base grows.
• Keep dragging past the point where the arcs stop crossing. The triangle disappears entirely — there is no angle you could swing those two sides to that would still let them meet.
• Press "Just barely", then nudge C one notch further with the arrow keys. The two arcs are still touching almost exactly at the base — the closest a triangle can get to failing while still existing.

Where this goes next

The same picture explains why a route through a third city is never shorter than flying direct: the direct distance is the base, and any two-leg path is exactly two sides that have to close a triangle around it, so their lengths must add up to at least as much.