Suppose that lines a b c through the vertices A B C of a tri

Suppose that lines a, b, c through the vertices A, B, C of a triangle meet at three points inside the triangle. Label them X = a middot c, Y = a middot b, Z = b middot c. Show that one of the two following arrangements must occur: (i) A * X * Y and B* Y* Z and C* Z* X (shown in diagram), or (ii) A * Y * X and B * Z * Y and C * X * Z.

Solution

Hilbert’s axioms of betweenness are then:

B1: If A B C, then A, B and C are distinct points on a line, and C B A also holds.
B2: Given two distinct points A and B, there exists a point C such that A B C.

According to given theorems,

The axiom A*X*Y, B*Y*Z, C*Z*X are in proper order and it satisfies the Hilbert betweeness axioms condition.Hence arrangement (1) must be occur for this Problem.