This article presents an effort to use Induction in geometry to prove Sylvester Gallois Theorum.
Theorem statement can be viewed here
For the proof, let us rephrase the theorem as follows –
(1) Given a finite set of points in Euclidian space such that any line passing through two of the points passes through at least one more of them, all points have to be collinear.
Let n be the finite number of points in the given set.
Given 3 points, such that a line containing two of them passes through one more of them, they are by definition, collinear.
Thus (1) is true for n=3.
Let (1) be true for a given k points set.
For an external point in the same space to pass through any line passing through two of the points in the k-points set in P(k), it has to pass through the line passing through all of them.
Thus the only way to expand the given set of k points to a set of k+1 number of points, is to include an external point which is collinear with the existing points.
Hence all k+1 points of the expanded set are collinear.
Thus if (1) is true for n=k, then it is true for n=k+1
Thus by principle of induction, (1) is true for all n>=3.