TITLE: Prove a line cannot intersect all 3 of the sides of a triangle at interior points
QUESTION [12 upvotes]: Here, a story is told of a competitor in the Russian math olympiad of 1945. The boy did not solve a single problem, but received a prize for writing:

I spent much time trying to prove that a straight line can't intersect
  three sides of a triangle in their interior points but failed for, to
  my consternation, I realized that I have no notion of what a straight
  line is.

Assume the standard Euclidean geometry of the plane. What is the simplest way to prove this statement? Can it be done without analytic geometry?
As noted in the quote, a lot of this has to do with how to rigorously define. A related question is: how is Euclidean geometry rigorously defined today, and in particular, what is a line? What is a point? A triangle?

REPLY [2 votes]: A line l divides the plane into two halves. Out of three points of the triangle A, B, C, two of them will definitely lie on one half (by the Pigeonhole principle and noting that none of them lie on the line itself because of the initial problem condition). Let these two points be A, B. Then the line l does not intersect the line AB of the triangle.