Answer to Question #134835 in Geometry for Jay

Question

Suppose that in a certain metric geometry satisfying Axioms D − 1 to D − 3, points A, B, C, D are collinear and
AB = 4 = AC ,AD = 6 ,BC = 8 ,BD = 9, CD = 1
(a) What betweenness relations follow?(Note: no ruler postulate so use
def. of between)
(b) Show that the triangle inequality is not satisfied.

Answer 1

Consider the following Metric Axioms.

D-1: Each pair of points A and B is associated with a unique real number, called the distance from A to B, denoted by AB.

D-2: For all points A and B, AB  0, with equality only when A = B.

D-3: For all points A and B, AB = BA.

Suppose A, B, C and D are collinear

Also, AB = 4 = AC, AD = 6, BC = 8, BD = 9, and CD = 1.

The objective is to show that there is a betweenness among the points.

AB = 4 = AC, BC = 8 confirms that B and C are on either sides of A and at equal distances.

Since BD = 9, and CD = 1, it trivially follows that D is close to C and farthest from B.

Use this information for betweenness as B – A – C – D.

Also, BC + CD = 9 = BD, BA + AC = 8 = BC confirms the details.