Synthetic and analytic techniques

Introduction:


Synthetic and analytic techniques can be used to prove and derive many geometric relationships. Additionally, a straight edge and compass can be used to construct a variety of well-known geometric relationships.

Task:

Construction: The midpoint of a line segment can be constructed.

Theorem: If the consecutive midpoints of the sides of a parallelogram are joined in order, then the quadrilateral formed from the midpoints is a parallelogram.

Write an essay (suggested length of 2–4 pages, including diagrams) in which you do the following:

A. Perform the construction given above using an unmarked straightedge and compass.
1. Describe each step of the construction.

B. Prove the theorem given above in Euclidean geometry using synthetic techniques.
1. Include each step of your proof.
2. Provide written justification for each step of your proof.

C. Prove the theorem given above in Euclidean geometry using analytic techniques.
1. Include each step of your proof.
2. Provide written justification for each step of your proof.

D. Compare the advantages and disadvantages of the synthetic and analytic techniques for proving the given theorem in Euclidean geometry.

E. When you use sources, include all in-text citations and references in APA format.