Question

this is #5 page 250, from The Mathematics That Every Secondary School Math Teacher Needs to Know. ISBN: 9781138228610

geometry), that lines are infinitely long? 5 In Euclidean geometry, one axiom is: If A, B, and C are three points lying on a line, then only one of them is between the other two. Show that if we work on a sphere and take our "lines" to be great circles, then this axiom fails. 6+ After doing

Answer 1

Axiom : If A, B and C are three points lying on a line, then only one of them is between the other two.

Now, we need to consider "lines" to be great circles.

Modified Axiom : If A, B and C are three points lying on a great circle, then only one of them is between the other two.

Our aim is to prove that the modified axiom fails. Note that a great circle is also a circle by it's definition. Let A, B and C be given points on a great circle. We will use the method of proof by contradiction.

Therefore by way of contradiction, assume that only one among {A, B, C} lies between the other two (i.e. the modified axiom is true). Without loss of generality, let B be between A and C. By modified axiom, the same property is not true for A and C individually. Now consider the point A. Now traverse along the great circle in the anticlockwise direction. When we complete 360⁰, we clearly reach back A. But since the points B and C are also on the great circle, we will reach those vertices clearly before reaching 360⁰. Without loss of genarality assume that we reach B before C. Now from A, traverse along the great circle in clockwise direction. Then we will reach C before B. Therefore, effectively from the point A, we reach B if we traverse in anticlockwise direction and we reach C if traverse in clockwise direction. Therefore the point A lies in between the points B and C, but this is a contradiction to our original assumption that the modified axiom is true.

Hence, our assumption was wrong and therefore the modified axiom fails, as desired.