Jump to Appendix − Converses to the Circle Theorems - The original theorem is used in the proof of. One of the basic axioms of geometry is that a line.
In geometry there is a theorem--Midsegment Theorem—that states:
The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to half the length of the third side.
To illustrate this theorem, consider ∆ABC in Figure 1.
Fig. 1
The midpoints of the segments ACand BCare points D and E, respectively. According to the Midsegment Theorem the segment DE is parallel to AB and its length is one-half the length of AB. This is true in any triangle and the proof can be found in any geometry textbooks employing parallelograms.
Vectors can also provide proofs to theorems in geometry. To prove the Midsegment Theorem using vectors, we need to modify Fig. 1 to what is shown in Figure 2. We wish to show that DE is parallel to AB and its length is one-half the length of AB.
Fig. 2
By construction D and E are midpoints of segments AC and BC, this means
By vector subtraction we have
Hence,
is equal to a scalar ½, times the vector , this implies that and are parallel, hence, DE is parallel to AB.
Now taking the lengths of the two vectors we obtain
This shows that the length of is one-half the length of , which means DE = (1/2)AB. But do we really need this higher-dimension math proof when a 2300-year old math can do the trick?
There are lots of vector proofs out there for geometry theorems but the most amazing thing about some of these vector proofs is their simplicity and elegance.
There are lots of vector proofs out there for geometry theorems but the most amazing thing about some of these vector proofs is their simplicity and elegance.