Consider two ‘postulates’ given below :
(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.
(ii) There exist at least three points that are not on the same line. Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.
In postulate (i) ‘in between A and B’ remains an undefined term which appeals to our geometric intuition.
The postulates are consistent. They do not contradict each other. Both of these postulates do not follow from Euclid’s postulates However, they follow from the axiom given below.
Given two distinct points, thre is a unique line that passes through them.
(i) Let A ---------- B be a straight line. There are an infinite number of points composing this line. Choose any except the two end-points A and B. This point lies between A and B.
(ii) If there are only two points, they can always be connected by a straight line (From Euclid’s postulate). Therefore, there have to be at least three points for one of them not to fall on the straight line between the other two.