Answer to Question #289 in Geometry for Kevin

Question

Can the points A, B and C belong to the same line, if AB > BC, AB > AC, and AB < AB + BC?

Answer 1

According to the second Euclid's axiom, one of the three points on a line locates between two others. According to the third Euclid's axiom the length of the segment is the sum of parts of length, on which it can be broken by any of its points. So, if A, B and C belongs to the same line, one of the statements must be true:

1) AB = BC + AC
2) AC = AB + BC
3) BC = AB + AC

The first statement is not true because AB < AB + BC (it’s the given information). The second statement is not true because AB > AC, so AB + BC > AC too, thus AC is not equal to AB+BC. The third statement is not true because AB > BC, so AB + AC > BC too, thus BC is not equal to AB+AC. So we can conclude that the points A, B and C can not belong to the same line.