Question

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.

Answer 1

Postulate (i):
"Given any two distinct points A and B, there exists a third point C which is in between A and B."

Undefined Terms:
This postulate implies the existence of a third point between two given points. However, it uses the term in between, which refers to the concept of betweenness. The concept of betweenness is not explicitly defined in Euclidean geometry. In Euclid's postulates, we have undefined terms like point, line, and straight line, but the notion of betweenness is implicitly considered.
 

Consistency with Euclid’s Postulates:
This postulate follows from Euclid’s postulates, specifically the notion of a line segment. Euclid’s first postulate states that "a straight line can be drawn from any one point to any other," which allows the existence of points between two given points. Thus, from Euclid's first postulate, a straight line joining two points can be divided into multiple points, one of which lies between the two given points.
In Euclidean geometry, the concept of points between two distinct points is a natural extension of the first postulate.
Therefore, this postulate is consistent with Euclid's postulates.
 

Postulate (ii):
"There exist at least three points that are not on the same line."

Undefined Terms:
This postulate refers to the concept of points not being on the same line, which implies the idea of non-collinearity of points. The term collinearity is also not explicitly defined in Euclidean geometry, but it can be understood as the property of points lying on the same straight line.
 

Consistency with Euclid’s Postulates:
This postulate is consistent with Euclid’s postulates, specifically the first and second postulates. The second postulate states that "any two points can be joined by a straight line." However, it does not imply that there must be only two points on a straight line, nor does it imply that points must always be collinear.
The existence of three non-collinear points is fundamental to defining a plane in Euclidean geometry. In fact, a triangle is formed by connecting any three non-collinear points, which is consistent with this postulate.
Therefore, this postulate does not contradict Euclid’s postulates and is in harmony with them.
 

Conclusion:

Both postulates (i) and (ii) are consistent with Euclid's postulates. The first postulate aligns with the concept of a straight line segment, and the second postulate supports the idea of non-collinear points, which is essential in Euclidean geometry for defining a plane and shapes like triangles.