A student writes: 'Triangle ABC has A(0, 0), B(6, 0), and a third vertex C for which AC = 2 and BC = 3.' Explain why no such point C can exist.

AB = |6 - 0| = 6. Every point at distance 2 from A lies on a circle of radius 2 centred at A, and every point at distance 3 from B lies on a circle of radius 3 centred at B. Those circles meet only if the distance between the centres is at most the sum of the radii: here $2+3=5<6$, so they are too far apart and there is no common point. This is the triangle inequality, AC + BC > AB. Testing $C$ = (2, 0) gives AC = 2 but BC = 4, not 3.