What Simple Theorems Can Be Proved Using Hilbert's Axioms of Geometry?

[Yankel]

Hello all,

I am looking for simple theorems that can be proved by using Hilbert's axioms of Geometry only. For example, such a theorem can be "two lines intersect in a single point". I am looking for more examples that can be proved (with a short proof) using these axioms. Can you think of such theorems?

Thank you !

 

[jvicens]

Hi there,

I am happy to help you with finding simple theorems that can be proved using Hilbert's axioms of Geometry. Here are a few examples:

1. Two distinct points always determine a unique line.
Proof: According to Hilbert's first axiom, two points determine a line. Now, if we assume that there are two distinct lines passing through these points, then by Hilbert's second axiom, these lines must intersect at a point. But this contradicts the given assumption that the points are distinct. Therefore, there can only be one line passing through the two points.

2. The sum of the angles in a triangle is 180 degrees.
Proof: Consider a triangle ABC. By Hilbert's axioms, we can construct a line parallel to BC passing through point A. Let this line intersect BC at point D. Now, by the parallel postulate, we know that the sum of the interior angles of a triangle and the exterior angle formed by the parallel line is 180 degrees. Therefore, the sum of angles A and B must be equal to angle ADC, which is 180 degrees. Similarly, the sum of angles B and C must be equal to angle ADB, which is also 180 degrees. Hence, the sum of the angles in triangle ABC must be 180 degrees.

3. The opposite sides of a parallelogram are equal.
Proof: Let ABCD be a parallelogram. By Hilbert's axioms, we can construct a line parallel to AD passing through point B, and a line parallel to AB passing through point D. Let these lines intersect at point E. By the parallel postulate, we know that angles AEB and DEC are equal. Also, angles AEB and ADB are equal as they are alternate interior angles. Similarly, angles DEC and CDB are equal. Therefore, triangles AEB and CDB are congruent by the ASA postulate. Hence, their corresponding sides, AB and CD, are equal.

I hope these examples help you in your search for simple theorems that can be proved using Hilbert's axioms of Geometry. Good luck!