- #1

- 21

- 3

## Main Question or Discussion Point

Before looking at the proof of basic theorems in Euclidean plane geometry, I feel that I should draw pictures or use other physical objects to have some idea why the theorem must be true. After all, I should not just plainly play the "game of logic". And, it is from such observations in real life that result in the list of axioms. But I have great difficulty in doing so. In other words, I have great difficulty in understanding the natural behaviour in the real world that is described in the theorem. For example, below are two basic theorems from Euclidean plane geometry.

1. Given two lines cut by a transversal. If a pair of corresponding angles are congruent, then the lines are parallel.

2. (Basic proportionality theorem) If a line parallel to one side of a triangle intersects the other two sides in distinct points, then it cuts off segments which are proportional to these sides.

(and its converse)

Another example, 3. the sequence of 1/n (n is a positive integer) converges to 0. We can use the axiom of completeness to prove the Archimedean Property of R, which then can be used to prove the above statement. Basically, we set up the axiom so that statement 3 is true. In other words, we want 1/n to converge to 0. Again, I have difficulty visualizing that any positive real number must be smaller than some 1/n using physical objects.

What would you do for these 3 theorems? Feel free to use other basic theorems as examples in addition to these 3 theorems.

What should I do to overcome this problem?

Edit for further clarification, I'm asking a "non-mathematics" question. I'm not asking about proof technique, or how to prove the 3 theorems. I'm asking about the process before even thinking about a mathematical proof of the theorem.

1. Given two lines cut by a transversal. If a pair of corresponding angles are congruent, then the lines are parallel.

2. (Basic proportionality theorem) If a line parallel to one side of a triangle intersects the other two sides in distinct points, then it cuts off segments which are proportional to these sides.

(and its converse)

Another example, 3. the sequence of 1/n (n is a positive integer) converges to 0. We can use the axiom of completeness to prove the Archimedean Property of R, which then can be used to prove the above statement. Basically, we set up the axiom so that statement 3 is true. In other words, we want 1/n to converge to 0. Again, I have difficulty visualizing that any positive real number must be smaller than some 1/n using physical objects.

What would you do for these 3 theorems? Feel free to use other basic theorems as examples in addition to these 3 theorems.

What should I do to overcome this problem?

Edit for further clarification, I'm asking a "non-mathematics" question. I'm not asking about proof technique, or how to prove the 3 theorems. I'm asking about the process before even thinking about a mathematical proof of the theorem.

Last edited: