Axioms for Desargues configurations Axiom DC1 There exists a

Axioms for Desargues\' configurations:

Axiom DC.1. There exists at least one ponit.

Axiom DC.2. Each point has at least on polar.

Axiom DC.3, Each line has at most on pole.

Axiom DC.4. two distinct points are on at most one line.

Axiom DC.5. There are exactly 3 distinct points on each line.

Axiom DC. 6. If line m does not contain point P, then there is a point on both m and any polar of P.

a.Prove the dual of axiom DC.3

Each point has at most one polar.

b. Prove the dual of axiom DC. 4.

two distinct lines are on at most one point.

Thank you for your help. I want to get a answer sooner.

Solution

Let P be an arbitrary point. By Axiom DC.2 Each point has at least on polar.

Assumme that P has a second polar P\'.

By Axiom DC.4 we have two distinct points are on at most one line.

and by Axiom DC.5. we have There are exactly 3 distinct points on each line.

hence there is point T on P\' but not on P.

Let T \' be a polar of T then by Axiom D6. we have If line m does not contain point P, then there is a point on both m and any polar of P.

so that P and T \' intersect . But since T on P\'. P is on T \' and so line T \' joins P to a point on P\' which is a contradiction to the definition of polar. Thus P has exactly one polar.

(b)

Now we prove that two distinct lines are on at most one point.

To prove this we assume , if possible that two given lines have two common points A and B

But any two distinct points are incident with just one line . This tells us that each line is determined by these two points . Thus these two lines coincide , contradicting to our assumption that they are distinct.

Hence the two distinct lines are on at most one point.