Number of Lines Passing Through a Given Point in the Plane

Question

How can one prove that infinite number of lines pass through a given point in plane, using Euclid's axioms (or Hilbert's, if necessary)?

Answer 1

Let us assume that we are in $\Bbb R^{2}. $

Draw a square containing the point P. Consider a vertex V of the square. By Euclid's axioms, there is a line passing through the vertex V and the point P.

Next consider a vertex T adjacent to V.Again, we can find a line passing through T and V.

Between any two points in $\Bbb R^{2}$ there exists another point $P_1$ on the line by virtue of denseness of reals.

Again join a line between $P_1$ and V.

As there are uncountable real numbers on any given interval on $R$, so there are uncountably many points between $P_1$ and V. Hence, you can draw uncountably many lines passing through P.

Answer 2

This is one of Hilbert's axioms:

• There exist at least two points on a line. There exist at least three points that do not lie on a line.

Hilbert's axioms are independent, hence if the axiom above (or one of its equivalents) were not included in the system, then a line could be a plane (i.e. a line would be a model of the plane geometry)!

So, given a point $P$ it's guaranteed that there is a line $l$ not passing through it. It can be proved that $l$ passes through infinitely many points $P_1,P_2, \ldots$ and the lines $PP_1, PP_2,\ldots$ are all different.