Lecture 1: Matching and Comparison

Matching and Comparison
Axioms of Matching the Segments
A matching is a technic relation. It is possible to call the shape matches another shape and it can be denoted by ?. For example, A-B matches C-D, it can be written by A-B ? C-D.

Axioms of Matching:
Ax1 (How to construct a segment). Let A-B be a segment and C be a point on the line m. Then for all the ray on m and it started point C, there exists one-point D such that A-B ? C-D.
Ax2 (Matching the segments is an equivalent relation). Matching the segments satisfies the following:
1. Reflexive: A-B ? A-B.
2. Symmetric: A ? B ? C ? D then C ? D ? A – B.
3. Transitive: A ? B ? C ? D and C ? D ? E – F then A ? B ? E – F.
Ax3 (Sum the Segments).
If
1. A-B-C,
2. D-E-F,
3. A-B ? D-F,
4. B-C ? E-F,
Then A-C ? D-F.
Theorem 1. (Subtract the Segments).
If
1. A-B-C,
2. D-E-F,
3. A? B ? D – E,
4. A-C ? D-F,
Then B-C ? E-F.
Proof. Suppose B-C does not match E-F. Based on Ax1, there exists a point G such that D-E-G and B ?C ? E – G. And since B-C does not match E-F and based on Ax2, E-G does not match E-F.
So G ? F….. (1)
Thus, A? B ? D – E and B ?C ? E – G. Using Ax3, A ?C ? D – G. But A-C ? D-F. Based on Ax2, D – G ? D-F.
Now, using Ax1 and Ax2 lead to G = F…..(2).
From (1) and (2), one can get contradiction (C!).
So, B-C ? E-F. ?